Class 10th Mathematics Tamilnadu Board Solution
Exercise 4.1- The rates for the entrance tickets at a water theme park are listed below: Write…
- There are 6 Higher Secondary Schools, 8 High Schools and 13 Primary Schools in a…
- Find the order of the following matrices. (i) (ccc 1&-1&5 -2&3&4) (ii) (7 8 9)…
- A matrix has 8 elements. What are the possible orders it can have?…
- Matrix consists of 30 elements. What are the possible orders it can have?…
- Construct a 2x2 matrix A=[aij] whose elements are given by (i) aij = ij (ii) aij…
- Construct a 3 x 2 matrix A=[aij] whose elements are given by (i) a_ij = i/j (ii)…
- If A = (lrrr 1&-1&3&2 5&-4&7&4 6&0&9&8) , (i) find the order of the matrix (ii)…
- If a = (ll 2&3 4&1 5&0) , then find the transpose of A.
- If a = (ccc 1&2&3 2&4&5 3&-5&6) , then verify that (aT)T= A.
Exercise 4.2- Find the values of x, y and z from the matrix equation (cc 5x+2 0&4z+6) = (rr…
- Solve for x and y if (2x+y x-3y) = (r 5 13)
- If a = (rr 2&3 -9&5) - (rr 1&5 7&-1) , then find the additive inverse of A.…
- Let a = (ll 3&2 5&1) b = (cc 8&-1 4&3) . Find the matrix C if C = 2A + B.…
- If a = (c 4-2 5-9) b = (rr 8&2 -1&-3) find 6A - 3B.
- Find a and b if a (2 3) + b (r -1 1) = (c 10 5) .
- Find X and Y if 2X + 3Y = (ll 2&3 4&0) and 3X + 2Y = (rr 2&-2 -1&5) .…
- Solve for x and y if (x^2 y^2) + 3 (c 2x -y) = (r -9 4)
- if a = (ll 3&2 5&1) , b = (lr 1-2 2&3) o = (ll 0&0 0&0) then Verify: (i) A + B =…
- If then verify that A + (B + C) = (A + B) + C.
- An electronic company records each type of entertainment device sold at three…
- The fees structure for one - day admission to a swimming pool is as follows:…
Exercise 4.3- Determine whether the product of the matrices is defined in each case. If so,…
- Find the product of the matrices, if exists, (i) (2-1) (5 4) (ii) (cc 3&-2 5&1)…
- A fruit vendor sells fruits from his shop. Selling prices of Apple, Mango and…
- Find the values of x and y if (ll 1&2 3&3) (ll x&0 0) = (ll x&0 9&0) .…
- If a = (ll 5&3 7&5) , x = (x y) c = (c -5 -11) and if AX = C, then find the…
- If a = (ll 1&-1 2&3) then show that A^2 = 4A + 5I2 = O.
- If a = (ll 3&2 4&0) b = (ll 3&0 3&2) then find AB and BA. Are they equal?…
- If a = (rrr -1&2&1 1&2&3) , b = (0 1 2) c = (21) verify (AB) C = A (BC).…
- If a = (ll 5&2 7&3) b = (rr 3&-1 -1&1) verify that (AB)T = BT AT.…
- Prove that a = (ll 5&2 7&3) b = (rr 3&-2 -7&5) are inverses to each other under…
- Solve (rr x&1) (rr 1&0 -2&-3) (x 5) = (0) .
- If a = (ll 3&3 7&6) , b = (ll 8&7 0&9) c = (rr 2&-3 4&6) , find (A + B)C and AC…
Exercise 4.4- Which one of the following statements is not true?A. A scalar matrix is a square…
- Matrix A-[aij]mxn is a square matrix ifA. m n B. m n C. m = 1 D. m = n…
- If (cc 3x+7&5 y+1&2-3x) = (cc 1 8&8) then the values of x and y respectively…
- If A = (1 -2 3) and b = (c -1 2 -3) then A + BA. (0 0 0) B. (0 0 0) C. (-14) D.…
- If a matrix is of order 2 × 3, then the number of elements in the matrix isA. 5…
- If (ll 8&4 x&8) = 4 (ll 2&1 1&2) then the value of x isA. 1 B. 2 C. 1/4 D. 4…
- If A is of order 3 × 4 and B is of order 4 × 3, then the order of BA isA. 3 × 3…
- If a x (ll 1&1 0&2) = (1 , 2) , then the order of A isA. 2 × 1 B. 2 × 2 C. 1 × 2…
- If A and B are square matrices such that AB = I and BA = I, then B isA. Unit…
- If (ll 1&2 2&1) (x y) = (2 4) , then the values of x and y respectively, areA.…
- If A = (cc 1&-2 -3&4) and A + B = O, then B isA. (rr 1&-2 -3&4) B. (rr -1&2…
- If A = (cc 4&-2 6&-3) , then A^2 isA. (ll 16&4 36&9) B. (8-4 12-6) C. (rr -4&2…
- A is of order m x n and B is of order p x q, addition of A and B is possible…
- If (rr a&3 1&2) (r 2 -1) = (5 0) , then the value of a isA. 8 B. 4 C. 2 D. 11…
- If A = (ll alpha & beta gamma & - alpha) is such that A^2 = I, thenA. 1 + α^2 +…
- If A = [aij]2x2 and aij = i + j, then A =A. (12 34) B. (ll 2&3 3&4) C. (23 45)…
- (cc -1&0 0&1) (ll a c) = (cc 1&0 0&-1) , then the values of a, b, c and d…
- If a = (ll 7&2 1&3) a+b = (cc -1&0 2&-4) , then the matrix B =A. (ll 1&0 0&1)…
- If (ccc 5&1) (r 2 -1 3) = (20) , then the value of x isA. 7 B. -7 C. 1/7 D. 0…
- Which one of the following is true for any two square matrices A and B of same…
- The rates for the entrance tickets at a water theme park are listed below: Write…
- There are 6 Higher Secondary Schools, 8 High Schools and 13 Primary Schools in a…
- Find the order of the following matrices. (i) (ccc 1&-1&5 -2&3&4) (ii) (7 8 9)…
- A matrix has 8 elements. What are the possible orders it can have?…
- Matrix consists of 30 elements. What are the possible orders it can have?…
- Construct a 2x2 matrix A=[aij] whose elements are given by (i) aij = ij (ii) aij…
- Construct a 3 x 2 matrix A=[aij] whose elements are given by (i) a_ij = i/j (ii)…
- If A = (lrrr 1&-1&3&2 5&-4&7&4 6&0&9&8) , (i) find the order of the matrix (ii)…
- If a = (ll 2&3 4&1 5&0) , then find the transpose of A.
- If a = (ccc 1&2&3 2&4&5 3&-5&6) , then verify that (aT)T= A.
- Find the values of x, y and z from the matrix equation (cc 5x+2 0&4z+6) = (rr…
- Solve for x and y if (2x+y x-3y) = (r 5 13)
- If a = (rr 2&3 -9&5) - (rr 1&5 7&-1) , then find the additive inverse of A.…
- Let a = (ll 3&2 5&1) b = (cc 8&-1 4&3) . Find the matrix C if C = 2A + B.…
- If a = (c 4-2 5-9) b = (rr 8&2 -1&-3) find 6A - 3B.
- Find a and b if a (2 3) + b (r -1 1) = (c 10 5) .
- Find X and Y if 2X + 3Y = (ll 2&3 4&0) and 3X + 2Y = (rr 2&-2 -1&5) .…
- Solve for x and y if (x^2 y^2) + 3 (c 2x -y) = (r -9 4)
- if a = (ll 3&2 5&1) , b = (lr 1-2 2&3) o = (ll 0&0 0&0) then Verify: (i) A + B =…
- If then verify that A + (B + C) = (A + B) + C.
- An electronic company records each type of entertainment device sold at three…
- The fees structure for one - day admission to a swimming pool is as follows:…
- Determine whether the product of the matrices is defined in each case. If so,…
- Find the product of the matrices, if exists, (i) (2-1) (5 4) (ii) (cc 3&-2 5&1)…
- A fruit vendor sells fruits from his shop. Selling prices of Apple, Mango and…
- Find the values of x and y if (ll 1&2 3&3) (ll x&0 0) = (ll x&0 9&0) .…
- If a = (ll 5&3 7&5) , x = (x y) c = (c -5 -11) and if AX = C, then find the…
- If a = (ll 1&-1 2&3) then show that A^2 = 4A + 5I2 = O.
- If a = (ll 3&2 4&0) b = (ll 3&0 3&2) then find AB and BA. Are they equal?…
- If a = (rrr -1&2&1 1&2&3) , b = (0 1 2) c = (21) verify (AB) C = A (BC).…
- If a = (ll 5&2 7&3) b = (rr 3&-1 -1&1) verify that (AB)T = BT AT.…
- Prove that a = (ll 5&2 7&3) b = (rr 3&-2 -7&5) are inverses to each other under…
- Solve (rr x&1) (rr 1&0 -2&-3) (x 5) = (0) .
- If a = (ll 3&3 7&6) , b = (ll 8&7 0&9) c = (rr 2&-3 4&6) , find (A + B)C and AC…
- Which one of the following statements is not true?A. A scalar matrix is a square…
- Matrix A-[aij]mxn is a square matrix ifA. m n B. m n C. m = 1 D. m = n…
- If (cc 3x+7&5 y+1&2-3x) = (cc 1 8&8) then the values of x and y respectively…
- If A = (1 -2 3) and b = (c -1 2 -3) then A + BA. (0 0 0) B. (0 0 0) C. (-14) D.…
- If a matrix is of order 2 × 3, then the number of elements in the matrix isA. 5…
- If (ll 8&4 x&8) = 4 (ll 2&1 1&2) then the value of x isA. 1 B. 2 C. 1/4 D. 4…
- If A is of order 3 × 4 and B is of order 4 × 3, then the order of BA isA. 3 × 3…
- If a x (ll 1&1 0&2) = (1 , 2) , then the order of A isA. 2 × 1 B. 2 × 2 C. 1 × 2…
- If A and B are square matrices such that AB = I and BA = I, then B isA. Unit…
- If (ll 1&2 2&1) (x y) = (2 4) , then the values of x and y respectively, areA.…
- If A = (cc 1&-2 -3&4) and A + B = O, then B isA. (rr 1&-2 -3&4) B. (rr -1&2…
- If A = (cc 4&-2 6&-3) , then A^2 isA. (ll 16&4 36&9) B. (8-4 12-6) C. (rr -4&2…
- A is of order m x n and B is of order p x q, addition of A and B is possible…
- If (rr a&3 1&2) (r 2 -1) = (5 0) , then the value of a isA. 8 B. 4 C. 2 D. 11…
- If A = (ll alpha & beta gamma & - alpha) is such that A^2 = I, thenA. 1 + α^2 +…
- If A = [aij]2x2 and aij = i + j, then A =A. (12 34) B. (ll 2&3 3&4) C. (23 45)…
- (cc -1&0 0&1) (ll a c) = (cc 1&0 0&-1) , then the values of a, b, c and d…
- If a = (ll 7&2 1&3) a+b = (cc -1&0 2&-4) , then the matrix B =A. (ll 1&0 0&1)…
- If (ccc 5&1) (r 2 -1 3) = (20) , then the value of x isA. 7 B. -7 C. 1/7 D. 0…
- Which one of the following is true for any two square matrices A and B of same…
Exercise 4.1
Question 1.The rates for the entrance tickets at a water theme park are listed below:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARMAAABuCAIAAACLAGBEAAAAAXNSR0IArs4c6QAAERpJREFUeF7tXTuO1UoQffOWgdAIAWsgQEBAACxgAiAiQoIYQUJIAiIGiYhoIJgFAAEBIALWAAgh9sE7T2d0KMp2u93u62tfl4ORx7e/p6v6W6dr7/fv3//EEwgEAgMR+Hdg+AgeCAQC/yMQmhNyEAiUIBCaU4JaxAkEQnNCBgKBEgT2tEOwt7dXkkDECQRWhgBV5i/N2bF9NvQFO1ajiiIa4JSBKdxitlYGYMRaOwKhOWuXgKh/GQKhOWW49cT69OkThvXv37/np14QJT/xRYR8/fr1BIvtJ0+enD17djwgoTnjMVxdCpBvSLmqDUG8e/eu/r127Zr9tzo6SB8FsA+UoXouvQmG5vRCFAE8Anfu3Pnw4QO/Yqj89u3bu3fvFOjt27c3b97cKGooAPZ+9Ny/f3+j2bUmHprzPyzoI9GTESB0YHq33zH1Uj9n+9Su7xZuROyaISApJnvx4kVF4cxNjzp419mjnCqqDT9ollggc5cuXZKqfP78+fHjx1AeZoqS4++FCxeYbBc4toKtIwYjqnY5heRkj3+bgAN/fn/w4EFOav1hpLgIavV4B97za/Tx40cFPnPmDN4hCkDg6tWrr169IhT4mP/OBJkIXiBbrXii70R2/MlGsYERV0nhHUXir0gc3xGL5VQWKKTKmWjEfHCaiTBrQYQyoCLM1JawCzRbcvuOFFgqfuyqBSrrxhyWkNH1k4UdIOs7SijMC4RcuP3RljFQFpRggiiDakRRgBCwYdhsVmQt3AhACXbNoO9SA8TqUhsnWF2a0yVnVkatZGQCOwicZprSVaYDuCiaVoe7wBFKTFaaIM1JqA3DuwGBCCu6S9YBW0tzYrZ23Apojy9fvmDucevWLUzTX758iRkFJPL06dMMAQnWjOj58+dqvK7vCIDo+DU9Cz958mTrxECzOI6BfFAYlPPw8BDvL168QFH5HXMnFKl1ltKa+PiPly9ffv/+PSCiHJ87d47zNyxyDg4OesFBMIGJd1sefEeVr1+/nihkwTpH7Ti+7kwhNOcYSYgCVr0QRwgBpuloTmjRlStXBLSmSezS3rx5w5+6vuMnTmPSe6C/fv1qtiXm9xBEZsQxR8/Dhw+hJJjN47vEC2KhYQHfJ9hrOn/+PCECblRp/EW+tq9JgONE/9mzZ1bZgNigFU6OMtRf+wlxZJ850C8l2KAacUyXGqBp8a+m2m7mjX/Z9l3f7QwBaXZNrPGTy1GLB8kWJyFcz/DhKGQngTb99PywYnNTXlUwImZVIg2OIuKlOd1CLVyXpJKn1zmtwSxcSDnWOT0qPEhzkJZVldZVh+3b7Pq1+d1Ft7sOrtCKS8mj5thxhtMhqzl2z0BzeptOTtc2FJxmmiywvlPDm8v6VtDcQMrauYUKIzbzHbrOQQpsDj52VyYHqGZ78UtYfNqWXcY7JkVYY2i6WFZoLCfYWcQzCAHhFpozCLdZBEbjoRPVmUlZmUJzRuIWOwRlAG4tFlfhI9Vma6XfoYxjzNmhxhxSlRhzhqD1J6xwizGnDMCItXYEQnPWLgFR/zIEQnPKcNtgLJwD9hJIsNrBtIHmlWt+ZOWZBiEH0sEwVjwaK9gd32gUYLHR9HWW0mqAWJa1s5jsOrvQGUVZLtyPLo7LiPaQBKnlHC9aG7yRuTM68tVJlzsjoibwfAwP2qhKMwm3sPgc24JdR9pl6drGtilYa259t+aVQ7OrpTlWNHuVp67moJfpUoamnlj77qFY2fChOT3o6URcXRePzPUwPo/z3UfG5dNlaiCBs+VAYGdykhhz2Ov3CmtXPatrDgpvC2NhkXxb61V9tHVUaS2wXeqB1KxpRXrM4bCTsFvPVKfQnCzNSbSZRNyNOdbuxlmv9VJoxG5oLVzrcNQ1RvXKQXXN6WI6EAT2IM0xxzIOHEfDDq3N6iCpRK/R7IOQQuvHXqBcgNCcLM3pCmQ5Hk5z3L+yv8yh0CCu7URt7tY21H3vipKuYS3N0diS6M4FgtMc9686Gg7arcOyKkUmVWsdu4hxaWXLVCHhFntrdlrR8y5KSZqRK6oMwmsKkUOh+fr1qyuBWDr4PtJQbUA9hwSVcTdM6Ww8sZotCK0Jg0ZOYMUnB3uCdpn8bm8LUQqt7AyGR0ToXprhM6SK7WFDc7IwJC1e3apb3rgkXO9LZlsOhaa5GU3iChKcp9qo4ige6DriBeHlxo0bGjTs8qYJtxtbSPVBxbWSRFLNWK2MQA1Wm1YblCc0J0tzGAh0Lr7Y/hUtbVlT4GliRNIX9JfsMq1WdEkSkmp2pRAg8O267mGCvJ44cWJAHTYWFNMnVFzjgyhugEIDL7VCdSTF1d04hQBQPKWzv7/fWmTUujlEQ2G4a9J60gXOb+9B2QB4NL1DnMyp3lKCjamR44q4bTRHVhPcRMYddGg+o2BdGw+JJWyTpTO3vTUuwTlC4l29A5lkdneEOAgEu2nJpUsrgE2pc3trCkBFbe4f1N1bC4vPAb3MBEG5KsgkzYNyDDJz2W1jO2DxiaHpx48flomdaCAMfdClfGy7kgp+zgRaUJJFPmsNExIsqdHRlmQDJxY7wWzLpypxTpipZglI/+AWs7W5TT45vUmXilsUZfvRTHnMVHY+iHVRuF0JcyDNrJRwC89TZV12xFovAux3Yp2zUgnYjdna9I0XzLbpMY8cdwqBOM/ZqeaMykyGwKSak+P0JxwwTdb2kdEYBGpqDh0tVCQqWiccYyo5fVzg4O53lc2b+052J5+K0E1f5fwcnY8Ta5mW8A5iHU7l57W5kNU0hx6IdF/45ko8/5RpdGjLiT5FR+awGZG9CWxMYLFCIwNssOJ8pv71x3PFy5qr0cwMdQcC4iNYWx4gBty4cQwkaxrRFONT6zyHLLym0Yo1xEAhdVJhrfGttbkzNVe90jbnrZvxWzmyQKa8iF028I6NaBlgjrpII9/Mg4WRwbYCjrVOajao48ZZcCwytdidZQAKt2pjDkzr4TyDnYdzIgmPACxl2sTYKT9NzSV/mQYpxT1IlYgYbdD87hpBGBoicZUfRr5oe44toB6cOnVKWaMrlRfBKuWZcyIiEWiOirrbwQTI0LMIbUZlHE0kieoWnzqaA1UBEJQYiDucz7BKnMLdu3dvizWcLGu0+gS0kMmqs7mMICfq75c7R62jOVCV27dvE2uMPDB9t/P1RQwXIwWF/SKYJFzrYwSmc6X1rFvKAMTMAn3u1geQgsJX0BwIB6QE6zkKDZl9R0dHKs0apMcS17iK5TwT3+HKilMOAgJ2irgrmMfC2ldAYREM57UFrbjoKGLvoO6WcgNk6PkLGAIxsXqIJFHd4lNBc6AkjhGO9QzIWKgV52/SIn7kAzjk3/jp06etEGCm22QvbRGssqzZ8KojnK7JFRxkBaMTRYFu2LYuEGV1HBQL+8vqR0gj5fIYdQcCXCQjAJBRPwLEgBtzAZLqegblWznw+L01FMjtCNkrFBJOf1QTa/nb6rMJIZeyt6adU9ebqLLNXkY/jbF9HrpThEyHRqkVvvWqLbvtRkAcKb31cqlaRcpPR7iFxWflnmgpyYXFZ1lLhcVnGW4RKxA4RqDCOiewDARWiEAw21bY6FHlUQgEs20UfEuPHOucshaMdU4ZbhErEIh1TshAIDACgbE7BLw2tpVYAjsunnPlhBlRhYgaCGwBgQGao9vBaWXTek/2FmowpywtRA6fYLbZhiIJko81zrLf8ZNlAc6N2Zbrsw32Ds1DcWwyOEcOrWexiTD23tT8c9zMkDruzQw/MpjlJtESQjYBqKb1vqR36+4ix/XFyBLa6BODY7O25BHaE+jXLnmgawPZZ/TeR1cRKJeUSps15nDS5a7T594cH1jjNbsQzdbcqGAZs7L244xOriPYYbdya3mZgUjIjpy83RFImNBgj0aKNKMW1eLRo0cy2AMjBTJBW/LlWg0PxRzWaJIlXnLfaxMMxIAbMwKSIjgNzbpi+CzNwdX9IhG05g1rPKqmu5q+GZh3utrO2IZROhAjcmsZEkjBFlurKeobv8NMeybzRut5gqJAMlYw25wYWKDYuVgeiozuxTlfMLOt12BZVDZcEJ4OfHh4CO1yrEkhq3TwBSaxCkl3EYjLkBis5ZQG7z9//qzYl1RJCra9GEy6qlklix1IBCoBRpN1q6rLBtAtwlZaDnlmWNmsMafuhQn5RDeSw/jgfYbwtRYJcEFzxl/+vZT6lpWTzgUSLFosgZwfuLKMNhQrS3Mwklhqzcii9E5qlb7zM7MIWYSSY2ZrixrMttZJO73fJJyrifO3YGYbpkZYUbiRByJSoEJYHYprzVsKuhIBKxvjtdY2eJnz2I1acD8D0uAc2gSzzTUx2pH+S9xsFutVNTGZbXCAx7jLZrZZahEq0+qn2zpt1g6j25WWjy56PiOBqXXn2ukVdwtsFvg3sa+NQm5ud7KZsvU9xva2+/iSnmC2ucvogIyVAQHl6JLBbCsYqAqjhFFjArgAp0yqwuKzDLeIFQgcI5C1QxBoBQKBgEMgmG0hEoHAMASC2TYMrx0LHeucsgaNdU4ZbhErEIh1TshAIDACgVnsEHRZVY+oV0QNBDaLQLnmkBdgn3yzms3WaXupWzSC2dbbDrS6sKYhC/LZlstsc0fmjrnFo/3JvCZlWgag5TJDVgmG6utC12C29UJKiKwJiL1U2b4jqRky2wo1x7nXasKk/sa5LrMm5boqugkfo9ubhfGv4robh7saaWLNccWQm7Hw2dZsIKqNM55ahc82Oh4T98iNy1i3SLjBuLDDcS8HzmoRSE42LrgcbANnUtk7K5g+AB0TBLOtFXmyG5vGvqvw2QYrV9Qc1qya2VuvsfhJwg2Te8tQSHPguDZQXHkTYQPY8Wp6ZcjMkYRwEraC2dYEjVRfSEg+TSsT+YmDle8QWF9L1CI7BEmjMG7kVwnsTmtIu7+/n6Ah5Cc7ZUgQTjgwQnlmQvOesvq9eZFbLieh5MnXpU72lqFKgHLNsdlDi7CG096au5qkl4ytpJyqOEWqUuHJEsECj+5yg9lmMVfPwv6Fk3NKyLJ8thXuEKC2dpnudkKAgv2VmwSOgdPK5KFFkOK696F7d4jeu8NTMQD6Dt1M4gCJW6O6cF7d3lpzEuX8jdlJVyt3rUtzbMrNvbVBgj6x5ljvdFyV2dKq3w1mm4XFERMthuGzbbL5kc8ojBoT0Ac4ZXIZFp9luEWsQOAYgTo7BAFnILA2BEJz1tbiUd86CITm1MExUlkbAqE5a2vxqG8dBCprDq3EW+kGiZ/qVCVSCQQmRKBcc+SHg4Y2zQs46a5jwrpEVoHAdAgUag681sCO0x5p4V8MNbBxxMdWY77ET9NVN3IKBGohIOlHgpkn9NY5mYvCM2DYAbgDdfyrn2itZMuv02JrDa0zeLr40t2z7pr2RJnza5RZ8V0KFuCUtaZwKxlz6GksrboYYax9jbO3l5sUqtPBwQH+YiEE+2LS3fAd71ov4RJ3mAPyu72mvVb3EekEAkMRKNGcWvcNyCsbZ3dwZyfXf9A0jEtHR0esD8YcuouwXgSHVjXCBwIVESjRnFqcJLh2cPQvS5VbHDOnYqtEUvNHoERzMHGCiI+sG/YYoDnO8ZAzj50/a3okCBF9uQiUaA69KDvPU9iAdrO4BKOT7FHngw1uhkAPVCIgVAancrmCtfMlL9EcgIIlPtYk9nox3DfgZnFQMKxPGEau1wgohizrA5SKhPDYABDPFjsEZFPGEwjMEIG/fBmQkrkzT1BQEk0Z4JTJefBzynCLWIHAMQKFs7XALxBYOQLheWrlAhDVH4wAFzV/NGdwAhEhEFgxAjFbW3HjR9VHIBCaMwK8iLpiBEJzVtz4UfURCPwHjVQnqsbQJgwAAAAASUVORK5CYII=)
Write down the matrices for the rates of entrance tickets for adults, children and senior citizens. Also find the dimensions of the matrices.
Answer:The given table can be expressed as a matrix where each column denotes the week and weekend rates for Adult, Children and Senior Citizen.
![](data:image/png;base64,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)
The dimension of above matrix is 3 × 2.
Also, the same information can also be expressed as a matrix where each row denotes the week and weekend rates for Adult, Children and Senior Citizen.
![](data:image/png;base64,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)
The dimension of above matrix is 2 × 3.
Question 2.There are 6 Higher Secondary Schools, 8 High Schools and 13 Primary Schools in a town. Represent these data in the form of 3 × 1 and 1 × 3 matrices.
Answer:Representing the given information in a 3 × 1 matrix, we get,
![](data:image/png;base64,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)
Also, representing the given information in a 1 × 3 matrix, we get,
⇒ (6 8 13)
Question 3.Find the order of the following matrices.
(i) ![](data:image/png;base64,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)
(ii) ![](data:image/png;base64,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)
(iii)![](data:image/png;base64,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)
(iv) (3 4 5)
(v) ![](data:image/png;base64,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)
Answer:(i) Order is 2 × 3
(ii) Order is 3 × 1
(iii) Order is 3 × 3
(iv) Order is 1 × 3
(v) Order is 4 × 2
Question 4.A matrix has 8 elements. What are the possible orders it can have?
Answer:Since there are 8 elements, we can make the multiples of 8, which are:- 1, 8, 2, 4
Therefore, the possible orders of a matrix are 1×8; 8×1; 2×4 and 4×2.
Question 5.Matrix consists of 30 elements. What are the possible orders it can have?
Answer:Since there are 30 elements, we can make the multiples of 30, which are:- 1×30; 30×1; 2×15; 15×2; 3×10; 10×3; 5×6 and 6×5
Therefore, the possible orders of a matrix are 1×30; 30×1; 2×15; 15×2; 3×10; 10×3; 5×6 and 6×5
Question 6.Construct a 2x2 matrix A=[aij] whose elements are given by
(i) aij = ij
(ii) aij = 2i – j
(iii) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGAAAAA7CAYAAACE0Y8MAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAALqSURBVHja7Zq9UcMwFMc1AAtwl0fHAIlgheAiA5D46DgqTg0DiHRpWICOkgkoYIJsQJENsgMgGyVG+CuyJSv2v3hFkrtIej+9Tz22XC4ZpDuBEgAAACAAAAC9kwWnJ8ZoGwk5dr3WanV3Eo3YB6PJ+kbKUwAAAAgAAABksACkmJ9f8bNnYvTpIwY0Wa+Hyo/GnLENY+zLRxCW8ua0yXq9tIBdVuI7CwKAbBrKXnwBaLLeYAEoV0WMbVPXUSXVigUAWAAAAAAABJKKZlJDHj/OAMBzHWAGV5rePwBAzyUFwDfohnYJYBR93K1WJwDg0edfxuJaxxybeANlWoqY8ttdrKHJ2jbb+tfTUF29bAVIUSyhcHfyR/lpQ4lv5kKeq+/iiKTPVA4AFIBMIHmM+azNNO5+Sg/1ei9+UsegAJiuSMSX19rHDUUZQQDQiie+eFKBJjQAdS0oZCkEYI5yFLkgM+1qkobBAswSni9e9HeIAR4B5Cm7bQCQOhbw289QL/0XRG/JjfyxCiHmY52eKpczYbQrPszPEMsgvK/uaMun4nbf1k1L7r/1QhorfD+ADyINrRJYQMcAkltP9BoSAGW9ymqLMjqXYg7nqr1oj9F7AHovqm7JtoK7BJDXVegtAKXoULO0pGWdSetbA5D2wlW+/psxdQQgqSssHkG8WuePJZRdkFYW6gKAi+pbAW3bmtJaqvipskUAk/dD30ObV9X5B7OdVrYFULaerq+K/reZ4jMTCD77QGVBrsl0tA2AqunoqoB8lLnz7tAFAc62OLS1gKr1yiYmjhNATuMw/9B+AFStV/bbYAEc0pmt414BABYQTgwIE0CPYkCdUj8UAL3MgvY+PF/BttPRLgDovTipA0JwQ//eqxtMR7sA4KUS7tQKAukF5fl5b72gzg9e0XH0tg/jMtS5IEcP4JDee/s3Pn86Wu0paZNT9OrkPSBEUQNl2Rcx5+uVTEebr3ODAHCsAiUAAABAAGC48g17mcZAd90s8wAAAABJRU5ErkJggg==)
Answer:(i) Since aij = i × j, and the general of matrix is:
![](data:image/png;base64,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)
On substituting the values, we get,
![](data:image/png;base64,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)
(ii) Since the general of matrix is:
![](data:image/png;base64,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)
On substituting the values, we get,
![](data:image/png;base64,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)
(iii) Since the general of matrix is:
![](data:image/png;base64,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)
On substituting the values, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGUAAABWCAMAAADhTKj+AAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAttv/tv//25A627Zm27aQ2////7Zm/9uQ/9u2//+2///b8tnicAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADUklEQVRoQ+1Z61LVMBBOULEqHsQL1qOFCqc9ff8nNGmbdJM2eyvo6NgfzADJfnvP5osx/78/7oGHzz/EOrS3si39J/vi3m3prgX7uiu/hf91h7t2RBm+fWXvGmr+2iB0QjHdW7Z+7eUjW6MMxbSvmVvPH+WBNLMthr2ZrQ7UOqBwjWFrk7gmojC3a6JioseMaTiROd/IE8xZFW0x3RtGWE9TSkq/9iLIHuoP5Oah5hiciRmOlbX25Zfpz4tZRbSuUjkMymOIYChC+YN2B1gxHK19f0eJ3Po/qeli7VBfPva1KhVIly2pcvJJc7KaKJEua2KjbLwZ5xtFxrnCxF2wSPUO8yiK9uxcEItnM6yLhyaU6af46yq0MGEFIyiNq8HpG5WOv9lZH8LRi0ODx1RxGdDchGGYo0/3pM2KwQID/dn6JNZlMrEPCvV2KYPvhgws/EvwnRvcjGUPmgxze9HwN3ieFzN6+PnO2lfg8MLqDE8NpGqai++mr4GKmCRlpc9H/Zgu4UO80lWq6pglJ80rgUy9sAelPx7gWYSgnKyuBn1OZfmIiNKjOJ88VLDdI6IQMxltORGMOH8fSlLvz4AyzXlJz+uq4sGkLn1fkH1y4KWl112B/NOimN7Pltew52UFDo9oNco6N0YUn9/JoTque2oUCN5ehP7zd6L8bo8l0c/mVQ3rEWclWC9JJqe1H1gPRnNZL0GqMkGJrIcSpXhU5X2MvIeUFUD6WN6ud6AIzpfnQcnN3IGCHCL5DLMHpTzZPSEK0qzyWU1vCzo/ZhokY7OobND5EcYsZT1EGH7mR+bHXaMSVAQVtGe4TKxFpyHxrajEouDnIUEH5NEpsSiEtsKsKrEoxE2c5GlSa0osCiFGxrsUWRTKJQvbw6iQIotCSYljwIr12EAtoZAeEQWmxKKQQkgCLTGpwKLQXZZeAXC2WRSGpqS10JhtFoUjgkXTB6hNFoXzUBAY+BXrwUjtcQmHgXeXjukKsGI9uCis1wQzGzx6TnOf5bkcPtMIe7Q3l/nKs7w5ZawHz2Xc965ZmxXrwULhmuKiEe4eKevBQuGa4nMx3DTF44bgVdTEF16C5F5ZJ3nhdT673WA9GB4TvVaPFbxiPRgo4iVr1kMs4h/a8AvCu0yXCqfDBAAAAABJRU5ErkJggg==)
Question 7.Construct a 3 x 2 matrix A=[aij] whose elements are given by
(i) ![](data:image/png;base64,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)
(ii) ![](data:image/png;base64,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)
(iii) ![](data:image/png;base64,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)
Answer:(i) Since the general of matrix is:
![](data:image/png;base64,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)
On substituting the values, we get,
![](data:image/png;base64,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)
(ii) Since the general of matrix is:
![](data:image/png;base64,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)
On substituting the values, we get,
![](data:image/png;base64,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)
(iii) Since the general of matrix is:
![](data:image/png;base64,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)
On substituting the values, we get,
![](data:image/png;base64,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)
Question 8.If A =
, (i) find the order of the matrix
(ii) write down the elements A24 and a32
(iii) in which row and column does the element 7 occur?
Answer:(i) Order of matrix is 3 × 4.
(ii) Since the general of matrix is:
![](data:image/png;base64,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)
⇒ a24 = 4 and a32 = 0
(iii) Element 7 occurs in 2nd row 3rd column
Question 9.If
, then find the transpose of A.
Answer:For the transpose of a matrix, we know that,
ATij = Aji, therefore the transpose of matrix A is:-
![](data:image/png;base64,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)
Question 10.If
, then verify that (aT)T= A.
Answer:For the transpose of a matrix, we know that,
ATij = Aji, therefore the transpose of matrix A is:-
![](data:image/png;base64,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)
Now, applying transpose on AT , we get,
![](data:image/png;base64,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)
∴ We see that (AT)T = A
The rates for the entrance tickets at a water theme park are listed below:
Write down the matrices for the rates of entrance tickets for adults, children and senior citizens. Also find the dimensions of the matrices.
Answer:
The given table can be expressed as a matrix where each column denotes the week and weekend rates for Adult, Children and Senior Citizen.
The dimension of above matrix is 3 × 2.
Also, the same information can also be expressed as a matrix where each row denotes the week and weekend rates for Adult, Children and Senior Citizen.
The dimension of above matrix is 2 × 3.
Question 2.
There are 6 Higher Secondary Schools, 8 High Schools and 13 Primary Schools in a town. Represent these data in the form of 3 × 1 and 1 × 3 matrices.
Answer:
Representing the given information in a 3 × 1 matrix, we get,
Also, representing the given information in a 1 × 3 matrix, we get,
⇒ (6 8 13)
Question 3.
Find the order of the following matrices.
(i)
(ii)
(iii)
(iv) (3 4 5)
(v)
Answer:
(i) Order is 2 × 3
(ii) Order is 3 × 1
(iii) Order is 3 × 3
(iv) Order is 1 × 3
(v) Order is 4 × 2
Question 4.
A matrix has 8 elements. What are the possible orders it can have?
Answer:
Since there are 8 elements, we can make the multiples of 8, which are:- 1, 8, 2, 4
Therefore, the possible orders of a matrix are 1×8; 8×1; 2×4 and 4×2.
Question 5.
Matrix consists of 30 elements. What are the possible orders it can have?
Answer:
Since there are 30 elements, we can make the multiples of 30, which are:- 1×30; 30×1; 2×15; 15×2; 3×10; 10×3; 5×6 and 6×5
Therefore, the possible orders of a matrix are 1×30; 30×1; 2×15; 15×2; 3×10; 10×3; 5×6 and 6×5
Question 6.
Construct a 2x2 matrix A=[aij] whose elements are given by
(i) aij = ij
(ii) aij = 2i – j
(iii)
Answer:
(i) Since aij = i × j, and the general of matrix is:
On substituting the values, we get,
(ii) Since the general of matrix is:
On substituting the values, we get,
(iii) Since the general of matrix is:
On substituting the values, we get,
Question 7.
Construct a 3 x 2 matrix A=[aij] whose elements are given by
(i)
(ii)
(iii)
Answer:
(i) Since the general of matrix is:
On substituting the values, we get,
(ii) Since the general of matrix is:
On substituting the values, we get,
(iii) Since the general of matrix is:
On substituting the values, we get,
Question 8.
If A = , (i) find the order of the matrix
(ii) write down the elements A24 and a32
(iii) in which row and column does the element 7 occur?
Answer:
(i) Order of matrix is 3 × 4.
(ii) Since the general of matrix is:
⇒ a24 = 4 and a32 = 0
(iii) Element 7 occurs in 2nd row 3rd column
Question 9.
If , then find the transpose of A.
Answer:
For the transpose of a matrix, we know that,
ATij = Aji, therefore the transpose of matrix A is:-
Question 10.
If , then verify that (aT)T= A.
Answer:
For the transpose of a matrix, we know that,
ATij = Aji, therefore the transpose of matrix A is:-
Now, applying transpose on AT , we get,
∴ We see that (AT)T = A
Exercise 4.2
Question 1.Find the values of x, y and z from the matrix equation ![](data:image/png;base64,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)
Answer:Since given is the matrix equation we would equate the right hand side elements with left hand side elements
⇒ 5x + 2 = 12 , y – 4 = – 8 and 4z + 6 = 2
⇒ 5x = 10 , y = – 4 and 4z = – 4
⇒ x = 2 , y = – 4 and z = – 1
Question 2.Solve for x and y if ![](data:image/png;base64,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)
Answer:Since given is the matrix equation we would equate the right hand side elements with left hand side elements
2x + y = 5 …1
and x – 3y = 13 ….2
Multiplying equation 2 by 2
we get (x – 3y = 13 ) 2
2x – 6y = 26 ….3
Subtracting equation 1 and 3
![](data:image/png;base64,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)
Or
Y = – 3
Substituting value of y in 1
2x – 3 = 5 ⇒ 2x = 8 ⇒ x = 4
Hence the solution is x = 4 and y = – 3
Question 3.If
, then find the additive inverse of A.
Answer:Let us first solve for the value of matrix A =
![](data:image/png;base64,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)
⇒ A = ![](data:image/png;base64,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)
The additive inverse of A = negative of the matrix
– A = additive inverse of A = ![](data:image/png;base64,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)
Question 4.Let
. Find the matrix C if C = 2A + B.
Answer:Given A =
and B = ![](data:image/png;base64,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)
We have to find matrix C where C = 2A + B
Now 2A = 2 ![](data:image/png;base64,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)
We would multiply each term of A with 2
2A = ![](data:image/png;base64,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)
2A + B =
= ![](data:image/png;base64,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)
C = ![](data:image/png;base64,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)
Question 5.If
find 6A – 3B.
Answer:Given A = ![](data:image/png;base64,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)
B = ![](data:image/png;base64,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)
6A = ![](data:image/png;base64,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)
3B = 3 ![](data:image/png;base64,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)
6A – 3B = ![](data:image/png;base64,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)
6A – 3B = ![](data:image/png;base64,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)
Question 6.Find a and b if a
.
Answer:![](data:image/png;base64,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)
⇒![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Equating both the sides of the matrix equation
We get
2a –b = 10 …..1
And 3a + b = 5 …..2
Adding equation 1 and 2
![](data:image/png;base64,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)
⇒ a = 3
Putting value of a in 1
2a –b = 10
⇒ 6 –b = 10 ⇒ b = – 4
Thus the required solutions are a = 3 and b = – 4
Question 7.Find X and Y if 2X + 3Y =
and 3X + 2Y =
.
Answer:Given 2x + 3y =
….1
and 3x + 2y =
...2
Adding 1 and 3 equations we get,
![](data:image/png;base64,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)
5x + 5y = ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAmCAMAAAC/D0ZhAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bQe0Z4gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABQElEQVRIS92WbVPCMAzHW1ApOJk8TqQ4hLXr9/+EJC3sPE122dk7D/M23Y88/JOg1F1aPdpzcZ+WrKv7pC21TgRnOJJf6PEHvLBa6zlbpLacJF9brhiSKw51JOEjAcnOGz67IaTm+ZyH1L7uVRZSqCB/INVbuqDy7BroSrRfk2IkWbLLTOrR+M0l0pOCWtGCCjsDFXxYS5UpWhWymCSo3CSc5GRfy9FNsCQk2QT/CekfZnfEFS1QQficav1Ir32HEk8NF5Ds6F35ip4WZ7pdIyHhXajp9TSMFNVkrxfkm7QGk/yuOJDyHEjCC1qcGdLMXLshqBMiTobOLrxtbt0QkmDX8RfWGfRJSek1bWln9JNQdPHeQUzkkbLow1/p/4eRIggoSl89kSW30UfX8GcCHrf+C907v4RhYRQi2mr39OgCyRAhIXhukREAAAAASUVORK5CYII=)
Dividing both the sides by 5
x + y =
…3
Now subtracting 1 from 2
![](data:image/png;base64,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)
X – y =
……4
Adding 3 and 4
![](data:image/jpeg;base64,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)
2x = ![](data:image/png;base64,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)
X = ![](data:image/png;base64,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)
Subtracting 4 from 3
![](data:image/png;base64,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)
2y = ![](data:image/png;base64,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)
Y = ![](data:image/png;base64,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)
Hence x = =
and y = ![](data:image/png;base64,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)
Question 8.Solve for x and y if ![](data:image/png;base64,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)
Answer:given
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Equating each element of the matrix equation with corresponding element
x2 + 6x = – 9
⇒ x2 + 6x + 9 = 0
⇒ x( x + 3) + 3(x + 3) = 0
⇒ (x + 3) (x + 3) = 0
x = – 3 , – 3
y2 – 3y = 4
⇒ y2 – 3y – 4 = 0
⇒ y( y – 4) + 1( y – 4) = 0
⇒ (y + 1) (y – 4) = 0
⇒ y = – 1 or 4
Hence the values of x = – 3, – 3 and y = – 1 , 4
Question 9.if
then
Verify: (i) A + B = B + A (ii) A + ( – A) = O = ( – A) + A.
Answer:9) given
A = ![](data:image/png;base64,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)
(i) To verify A + B = B + A
LHS:
A + B = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJEAAAAnCAMAAAA4lVp5AAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOmaQOma2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///b1lEYTwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACMklEQVRYR+2Y61KDMBCFk3opKkqrYm0qthbI+z+iuQG5bRKY4jgd8sNxBnLydbPZnAWhZSRHgB4fMb7be9+npw34LLDAaeuXM6e44m2B8WqPyOoTNSX7xzMIfkNNcfOFCMb4JfIjlR5qNpjNiA9XvC3WfBrhfyv87iUanrVFnEjo1fmhSiNyxBWRICEBjbOgTSbivy+JSCxsiA9EzS4/wFGWtDMRGeIdEd///AckqjOxofMQmeLarp0yMM60lPlzESJ+QOSQJ8kS1/PoDJ0kWop0vRCRtRG2uE5UZ8BJImu1nxeJkUVki0siGTiZ8u6o7hnQmb8xA5Ej3hGxLW1KvrI76ge23ZSMJvLXW0vfFVe71uwyjJ/9Z01lopeIllZYlR7levj2FS4n6okrrudRdLq7axBRXMp9Q6XEQhQI3jXGyK69qoaqYjo+k64xRvJO8p/+8RHqy+8/P2vfMWfs3CLzxajmJVa5A0/EOxefTJTQLTjNhpnZyjT5t39w8ck3bW/o4YRymo10Is3FpxMFOonuRrNfSSdiCp2LTyYSq0LuRoub3mz8AVGotxFUZrNhEz1lUE87NUbB3BT2z2w2TCL6wfpWoKedSNQZ+lCxNJoNzy0C+uwpedQb+mD51psNDxFcvydkdm/og0R6EHoi+SWCe9ZYjMZ8iRgMPUBkNxu2OBHOH+wgqyQXr62tGXqQKNBs8HO4ZZcI0Pcnu3i9zsimNfQlJdRsTLEOy5wlAkMEfgHv2UEHPcqofgAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
RHS = B + A
= ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, LHS = RHS hence proved
(ii) A + ( – A) = 0 = ( – A) + A
– A = ![](data:image/png;base64,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)
LHS:
= A + (– A)
= ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
RHS ( – A) + A =
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALUAAAAmCAMAAAC1fyp2AAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOmaQOma2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///b1lEYTwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACLklEQVRYR+2Y61LCMBBGE7xQFS2oFQ1WkJa8/yOaG0NpLvsFOoPDND8hc3Ky3WSTMDa2/xgBuX7k/O4LUJObOdjzAEvRUd6u5HzSExSTT9ZW/V9DkxD8jbXlzTcTnPMXapp2rBQd5e3KqTeY0D/V/J2yYKzTc1cC1hqcoqO8kLWxFSqCWNua+cHWJB3gRazb5WyFOe/nl2Odptt4JXlBa52As1/QuilMKuHWBB3hOWu9mmxzi3BThDPE6ygrm8+4teoco6u/IF40r7f0nqBlZeVWc5Y1i9IxXtS6Kcg9wSzaqcukPOsoHeMFrO03siuZavW9kt7q/qh1mg7ygtYqt9tK+1CteVAFSoo86wQd5YUypF0WnD8D0qYkuqroxVpW/Y9lx0rRU7xuBKN5TYW5/z9sDYLz92sQfNRttO7s/159cMXBPz2dEmp/D4nlNUi/QIYMEKELWNuqGd5D8mI9wPxTA46rkTyNxffrH+TieIh+ZqxJenZeN6o0+rfJSHpsFnZ2sDVBj/Ko2ujO5ciyaefcncPR0xNL0kFeKENw62a2qoe0RnnnWavPMag1yrsm66cCe3tCY2OXiI1QUyTp0LcL3Xblh3pRirw9ebUKGqVrnaKjUTj33pif12YG8VspFIWoNXpdgEbpxpo4akK84OukvgeCd3QX65zXyTTdWkO844IizH0UfOerkbfXLp+gZ/P27HahCjr0zif1vZjfviJlFKGfwssZe+x7bRH4A92ZQoF96WOKAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAXCAMAAABd273TAAAAAXNSR0IArs4c6QAAAEhQTFRFAAAAAAAAAAA6AABmADqQAGa2OjqQOmaQOpDbZgAAZjoAZrb/kDoAkGY6kNv/tmYAttv/tv//25A62////7Zm/9u2//+2///brd/baAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZ0lEQVQoU9WSSRKAIAwEE/cFBRXk/z/VqeCNUF6ZQ05NMl0FUc2JG/PgdINouiuY9lAJ3+xEnlcVsHh8z70G4AIATMtfsDVFAJnZ/ASkQ/YEpZKTanHCsKQJgULHd3FYmEfNobo/9ACwvwRwCb8bHgAAAABJRU5ErkJggg==)
Hence LHS = RHS = 0 proved
Question 10.If
then
verify that A + (B + C) = (A + B) + C.
Answer:10) given
A =
, B =
and C![](data:image/png;base64,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)
LHS = A + (B + C)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
RHS = (A + B) + C
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
LHS = RHS
Hence proved
Question 11.An electronic company records each type of entertainment device sold at three of their branch stores so that they can monitor their purchases of supplies. The sales in two weeks are shown in the following spreadsheets.
![](data:image/png;base64,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)
Find the sum of the items sold out in two weeks using matrix addition.
Answer:Here we consider the types of items sold along the column and the store in which they are stored along the rows
Thus week 1 and week 2 matrices can be written as
W1 =
and W2 = ![](data:image/png;base64,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)
The sum of the items sold in two weeks is the sum of the above two matrices, which is sum of each corresponding elements of the two matrices
W1 + W2 = ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
TV DVD video CD
⇒ The sum of items sold in two weeks = ![](data:image/png;base64,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)
Question 12.The fees structure for one – day admission to a swimming pool is as follows:
![](data:image/png;base64,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)
Write the matrix that represents the additional cost for non – membership.
Answer:Here we consider the type of member along the columns and the timings in the rows. Thus members and non – members matrices can be written as:
M =
and N =
respectively
The additional cost for non – members as compared to the members is the difference of the above two matrices, which is the difference of each element of the matrices to its corresponding element in the other matrix
N –M = ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
The additional cost of non – members as compared to the members is
![](data:image/png;base64,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)
Find the values of x, y and z from the matrix equation
Answer:
Since given is the matrix equation we would equate the right hand side elements with left hand side elements
⇒ 5x + 2 = 12 , y – 4 = – 8 and 4z + 6 = 2
⇒ 5x = 10 , y = – 4 and 4z = – 4
⇒ x = 2 , y = – 4 and z = – 1
Question 2.
Solve for x and y if
Answer:
Since given is the matrix equation we would equate the right hand side elements with left hand side elements
2x + y = 5 …1
and x – 3y = 13 ….2
Multiplying equation 2 by 2
we get (x – 3y = 13 ) 2
2x – 6y = 26 ….3
Subtracting equation 1 and 3
Or
Y = – 3
Substituting value of y in 1
2x – 3 = 5 ⇒ 2x = 8 ⇒ x = 4
Hence the solution is x = 4 and y = – 3
Question 3.
If , then find the additive inverse of A.
Answer:
Let us first solve for the value of matrix A =
⇒ A =
The additive inverse of A = negative of the matrix
– A = additive inverse of A =
Question 4.
Let . Find the matrix C if C = 2A + B.
Answer:
Given A = and B =
We have to find matrix C where C = 2A + B
Now 2A = 2
We would multiply each term of A with 2
2A =
2A + B = =
C =
Question 5.
If find 6A – 3B.
Answer:
Given A =
B =
6A =
3B = 3
6A – 3B =
6A – 3B =
Question 6.
Find a and b if a .
Answer:
⇒
⇒
Equating both the sides of the matrix equation
We get
2a –b = 10 …..1
And 3a + b = 5 …..2
Adding equation 1 and 2
⇒ a = 3
Putting value of a in 1
2a –b = 10
⇒ 6 –b = 10 ⇒ b = – 4
Thus the required solutions are a = 3 and b = – 4
Question 7.
Find X and Y if 2X + 3Y = and 3X + 2Y =
.
Answer:
Given 2x + 3y = ….1
and 3x + 2y = ...2
Adding 1 and 3 equations we get,
5x + 5y =
Dividing both the sides by 5
x + y = …3
Now subtracting 1 from 2
X – y = ……4
Adding 3 and 4
2x =
X =
Subtracting 4 from 3
2y =
Y =
Hence x = = and y =
Question 8.
Solve for x and y if
Answer:
given
⇒
⇒
Equating each element of the matrix equation with corresponding element
x2 + 6x = – 9
⇒ x2 + 6x + 9 = 0
⇒ x( x + 3) + 3(x + 3) = 0
⇒ (x + 3) (x + 3) = 0
x = – 3 , – 3
y2 – 3y = 4
⇒ y2 – 3y – 4 = 0
⇒ y( y – 4) + 1( y – 4) = 0
⇒ (y + 1) (y – 4) = 0
⇒ y = – 1 or 4
Hence the values of x = – 3, – 3 and y = – 1 , 4
Question 9.
if then
Verify: (i) A + B = B + A (ii) A + ( – A) = O = ( – A) + A.
Answer:
9) given
A =
(i) To verify A + B = B + A
LHS:
A + B =
RHS = B + A
=
Here, LHS = RHS hence proved
(ii) A + ( – A) = 0 = ( – A) + A
– A =
LHS:
= A + (– A)
=
RHS ( – A) + A =
Hence LHS = RHS = 0 proved
Question 10.
If then
verify that A + (B + C) = (A + B) + C.
Answer:
10) given
A = , B =
and C
LHS = A + (B + C)
⇒
⇒
⇒
RHS = (A + B) + C
⇒
⇒
⇒
LHS = RHS
Hence proved
Question 11.
An electronic company records each type of entertainment device sold at three of their branch stores so that they can monitor their purchases of supplies. The sales in two weeks are shown in the following spreadsheets.
Find the sum of the items sold out in two weeks using matrix addition.
Answer:
Here we consider the types of items sold along the column and the store in which they are stored along the rows
Thus week 1 and week 2 matrices can be written as
W1 = and W2 =
The sum of the items sold in two weeks is the sum of the above two matrices, which is sum of each corresponding elements of the two matrices
W1 + W2 =
⇒
TV DVD video CD
⇒ The sum of items sold in two weeks =
Question 12.
The fees structure for one – day admission to a swimming pool is as follows:
Write the matrix that represents the additional cost for non – membership.
Answer:
Here we consider the type of member along the columns and the timings in the rows. Thus members and non – members matrices can be written as:
M = and N =
respectively
The additional cost for non – members as compared to the members is the difference of the above two matrices, which is the difference of each element of the matrices to its corresponding element in the other matrix
N –M =
⇒
The additional cost of non – members as compared to the members is
Exercise 4.3
Question 1.Determine whether the product of the matrices is defined in each case. If so, state the
order of the product.
(i) AB, where A = [aij]4x3, B = [bij]3x2
(ii)PQ, where P = [pij]4x3, Q = [qij]4x3
(iii)MN, where M = [mij]3x1, N = [nij]1x5
(iv) RS, where R = [rij]2x2, S = [sij]2x2
Answer:(i) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.
⇒ Here A[aij]4 x 3 and B = [bij]3x2
⇒ Number of columns in A = 3
⇒ Number of rows in B = 3
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 4 × 3
(ii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.
⇒ Here P[pij]4 x 3 and Q = [qij]4x3
⇒ Number of columns in P = 3
⇒ Number of rows in Q = 4
Thus the product is not defined.
(iii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.
⇒ Here M[mij]3 x 1 and N = [nij]1x5
⇒ Number of columns in M = 1
⇒ Number of rows in N = 1
Thus the product is defined and the order if product is
Number of rows in M × Number of columns in N
∴ MN = 3 × 5
(iv) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.
⇒ Here R[rij]2 x 2 and S = [sij]2x2
⇒ Number of columns in R = 2
⇒ Number of rows in S = 2
Thus the product is defined and the order if product is
Number of rows in R × Number of columns in S
∴ RS = 2 × 2
Question 2.Find the product of the matrices, if exists,
(i)
(ii) ![](data:image/png;base64,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)
(iii)
(iv) ![](data:image/png;base64,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)
Answer:(i) ⇒ let A : [2 -1] ∴ A[aij]1 × 2
⇒ let B :
∴ B[bij]2 × 1
Number of columns in A = 2
Number of rows in B = 2
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 1 × 1
![](data:image/png;base64,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)
⇒ [2 × 5 + (-1) × 4]
⇒ [10-4]
⇒ [6]
(ii) ⇒ let A :
∴ A[aij]2 × 2
⇒ let B :
∴ B[bij]2 × 2
Number of columns in A = 2
Number of rows in B = 2
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 2 × 2
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
(iii) ⇒ let A :
∴ A[aij]2 × 3
⇒ let B :
∴ B[bij]3 × 2
Number of columns in A = 3
Number of rows in B = 3
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 2 × 2
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
(iv) let A :
∴ A[aij]2 × 1
let B : [2 7] ∴ B[bij]1 × 2
Number of columns in A = 1
Number of rows in B = 1
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 2 × 2
× [2 -7]
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Question 3.A fruit vendor sells fruits from his shop. Selling prices of Apple, Mango and Orange are ₹20, ₹10 and ₹5 each respectively. The sales in three days are given below
![](data:image/png;base64,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)
Write the matrix indicating the total amount collected on each day and hence find the total amount collected from selling of all three fruits combined.
Answer:⇒ Let the Sales matrix be A = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG4AAAA7CAMAAACtxk+NAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29v/2////7Zm/9uQ/9u2//+2///bl0v2TgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADFElEQVRYR+1Y21LbMBC1oAS3JTSmtOCmYKgh9fX/f697kS+KpdUODTOdjvWQB+Voj7S7Wh1vkvzfo8uMMZuKD/ny+E6HLYDkDI13GXE1KUzwTL835uo5xNs/pbRAQPWvn4y5oJ3PUIVLd28J+nxTtfn5Lz9fnV7/JjsCqjh7SNocrc9RAboacbUZ2F3WOr3jCQlVXAKgRAtzVICuwIN1GS5ZjGlaQtEyAsxRfjo8/xjRY77x0CIKVrX7LUTfQR3RfU4pvAzh38Uozn9+Nea6klHgG2O2oxVry6Hrf9xxeCW6PjdfquSQ7iJ0sM1DCs50bDl0dJBmNOSP3bDPTTW4yRthziUzbMraWtLRHza8O48vLV0JGxdQvBC37qAcugLNE4Qz2H8RiCQpIK4Cqs/RFlmYo1w6CBtfbiwzgUyB/VxWySsaElA9pkA7Jvhgy6Frb6GGYfJCEkPyYV75RgP/XTxEUO0eKiKkr2trGTs/xYlmV7oTORLNrM5cnan0wCJVShIvkugpUT+xcJOk0QFrjyiNqECzEJNEDyqYJoVfEWXg7cyw/galERbdb0QnSiM8Pb4/fyuN4BruiMiRM76wdBm+L1FpRA9QUBol9VVFdDHRk1CEoyhiCkuj7uaRHSRKIwotCoYYagrwKLPmmUkvMNCV9zFD/M7HUPyiB6VRzRlujKULyFoMGn8h2Mc6II3YBS5qce+cVMHdeQbJmSlVAqgCNMYxKkAnSSOba3gbBAGVlOiBGrYSlEZ0ocZPopA0gstpzyMKqI+Ycqjt5qjj00H8iE+SRuxvHAIKvx1h4MZmqPV59afnm2ZXZ77Jbf5FqzPf0ZmKftBU6iK9pcMtV59Q1whKqqIfBAYUAgpLF3edwtJI1Q/SCahm+0zySegaaftBSgFl6YLSSNkP0goopgtLI10/SC2g5nSerpGuH6QXUFE6UjygeyTRoxdQrjNJJ7nCj+g0/SCNgGJhL3WNNP0gurlMJwuogc4njegLStUPIjqFgBpAkzQaW99DVVT1g/BwcQHVY9fIfPjuSqMTlt9/1NQfmK2angZVT8wAAAAASUVORK5CYII=)
⇒ Selling price matrix B = ![](data:image/png;base64,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)
AB = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
These are the amounts earned on each day.
Total amount earned = 1750 + 1600 + 1650 = 5000 Rs
Question 4.Find the values of x and y if
.
Answer:= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Comparing with ![](data:image/png;base64,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)
3x = 9
∴ x = 3
And Y = 0
Question 5.If
and if AX = C, then find the values of x and y.
Answer:x = 2, y = - 5
⇒ AX = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Comparing with ![](data:image/png;base64,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)
5x + 3y = -5 ---1
7x + 5y = -11 ---2
Multiply 1 by 5 and multiply 2 by 3 and subtract,
![](data:image/png;base64,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)
x = 2 and y = -5
Question 6.If
then show that A2 = 4A + 5I2 = O.
Answer:⇒ A2 = AA = ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYEAAAArCAMAAACpd8QqAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZjpmZmaQZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAtpBmttv/tv//25A627Zm27aQ29u22////7Zm/9uQ/9u2//+2///b7wP6DgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFGElEQVR4Xu1b63rTMAxNBmPlsrGOawdllEGb5v0fkNiyY8uSYzl26caX/GAfLJLO0fEtyaFplmvpwNKB03Sg//HuZyTz7vpBVBOnkEZNpa4OChWbifC4btv2VmV6/PBd0hjZbf3m/R+azcQe3n4VVIIU/eNd214qZFNR/xhUiD6HV//rDRDaDo1/NTA8rtWfTdPdtS9ig9YryN7Wb8KO9hstKr5c7OF1WmyTYtt+abq1hhaNEmLPBnUSXtuLb023uVANgN4bBQ7XDzuswPEjCNKZn/ov9Db1rxTpnlHTj92B7HCxlRqTYns13LFrtcQoysULsduMaFhEQVXgxRPzCCEFFD/ctX5zqSQ4rm/wSA5uY5H2G9U3eo2xaDizlVCKPSgQnzoS7HmgKvAaRibfwmEF0r1OKDDEDwP1uA57KVHgsOIX+jEWLwhcJZQCAA+IbllhyehhsWeC0vUCGrkpIi3s7uEsklJA0fh9dxVuqBIF9nqZm5gDjZ6L48VU8lOMxHGUl4CASmT0i7vYMD1RIJcX20J19rnWXU0qMMTrzRpfAVm1o8Pldd0iDX/pkcWJaSWPrBv6WwoHwNFhEc8oA1WHF9/C5nElWoXULEwrwK6XzDwJGhX0klZyKbz1u0iBbFB0FZqTgmlh0+z1A0ByDuiZTEZdpVUI5WUquTmwdQuhXIHJjLFJHaafswqFc5tr4XCmWEkUsNtIsA5VUsDfB7hKowL6CLqHLVi8D0xmjC6r1fcBBgYsqXC4I3MAb586Wt0VHIZ2dJeVnxls7HENLdUXW8nuvvoE2m/17SgK9TEANZkxEKAZY0n6Ql4ssV49jHW2t94TWX+/GjbTl58dvO4TDP6D+QnNIrdBC4NTG3v09mPRyZ6vZB4pzI6oFYg8D0ixcw8pUVAVeDUssU618IY9C4Vjo+Tv3DOxny/ydOvfwqQQRE2BPg0oPBdjhzUWV7gKlXQ8jI0/O8G8kr8X8jJLoqZInAQUKpiJ8JQKDAsT927UwM15N+oYyqKmJagOCgsgeufrQk6qQFP9VfzMt++oRdVB4SVI+N1jDDqtAjXXtP8116LAuZVdFFgUOHcHzl1/mQOLAmfuQIWTURmD9BxwPoVUJZFXYXQJTGaTF+XTiOPL7RxCQtYcQfCmFfB8CpNNExooPJfARDpp0VgKaXwF54SQkDNHBJgFCng+BQguMlD4tod4OnpXagLi38uqDC+Ic+wcqgR1hMhKxQmlFdDUjE8BaJYYKCCD+eg+nQ4XzVNACLqCcyKHEKJtCQkVwLEFBgpV17oELIZIOqzTDAXSoHNtD9wcEBMKaRtGMgVCpPMNFOoLi3UJjE3l08XaI9VCADrb9sCsQkJCDG2zCPueuRg3ul9JPt+zRgNdwrgEXDkuXeI1clIHCeg8O0chIUpbc5DMAWa1lCgAWwbr2gKXwKQCsSU62Xlzgwj0HNvDXEIDrpC2WAHPpzDSm2uggATgEsCrUPBhiRaV9h7uE4EWrEIBrMiQEhCitAGoYA74PgUIKjBQ+C4B21IuHS2aJ4AMtECBwKLAfA1XgwmdFZn+cLQNobQCvk/BE2CmgcJ3CbgJRfwYtGieAELQ0bMQa+eAwUf8CM72ECfE0LaE0gr4PgWIKjFQDEfR0SVgMHDpaNE8BaSgy50TQkKUtlyBPOrP7O4KzolSxnYO2P/FVJrvmcVXcE6UMbb/i6ksy3OOrmDneM70nwL2s38feApNWDAsHTh3B/4C2kDY3Dq2/McAAAAASUVORK5CYII=)
= ![](data:image/png;base64,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)
⇒ 4A = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ 5I = ![](data:image/png;base64,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)
⇒ A2 - 4A + 5I2 = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADMAAAAmCAMAAABeZFMwAAAAAXNSR0IArs4c6QAAAEhQTFRFAAAAAAAAAAA6AABmADqQAGa2OjqQOmaQOpDbZgAAZjoAZrb/kDoAkGY6kNv/tmYAttv/tv//25A62////7Zm/9u2//+2///brd/baAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAzElEQVRIS+2U3Q6CMAyFW/FviqIC4/3fVOo6aUdNGhPxhl1wwemXdmfNAVj6DBfE/c3uWmoxIG6uMNTbtq+ruwW9tQYRj2NFDDuq60YOOjxbjNBiEExDLZgvOaFJhtoTQ9/ySG3OJNVmMjnNlv58w9j3yR3IK8sD6jw77EGaSvj2IJs/eC00xZBl9nWSnawpBvoT4sFwjSadNM349ntlijf12ab3YDHGlyHqTV0ZUniwZggnzm8z5JXxrgzJGc+L5swQvZa+DHGu8t/KniunGUcghsSPAAAAAElFTkSuQmCC)
Question 7.If
then find AB and BA. Are they equal?
Answer:⇒ AB = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ BA = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
AB not equal to BA
Question 8.If
verify (AB) C = A (BC).
Answer:⇒ AB = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ (AB)C = ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIEAAAAmCAMAAADUZwg0AAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmZmZmaQZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAttv/tv//25A627Zm27aQ29u22////7Zm/9uQ/9u2//+2///bok+hWgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACBUlEQVRYR+2XW1vDIAyGQZ2rx9V5mtWh3ewB/v8PlFC6EQxruNhduemePvTlI8nChxDzQBGoL7YTETH7tZSL9KxpAiywf0YEXUo5rNwVkwqU3Ii+vPxJCGUQhOjX0gGUlHIFIF0uB54uX5EC/TIs1Psn/FYwt5YftAIOQXT337Xfgi6xArVqkQJTLUCCLh+i9dqUAi4hoaC9a7ACYarr5hiiowyVyAKbQCvQT1sRKQAJv+tlE4WgK+gk8AmkAlPZlFgFNaKbStow4OFmEoNPEKSC1hamG1MKTOULNxbBJthSTlSiiwHiuixEQVD/shJ+wiFkKBgrMcxDDYJaOg9WSqKWo0ymY4A7mhMQ9AvYa3djo2RUUsE0ASjjrLgfCJvJMA392yC980/XkYZiSSlgEIT5LCzh6t1tDnckus+d8+2sYM4C1NdcB3MMiDrY3dqWCJ3q1DjpVFkEoI9OFVdifbERYpfst17WKafKIxydalQH/thXwVmc6VSZhKRTHb8P7EemU80g0Kdza28CYheZ5Synyick/MHeHpqUR2I7VcEm0Aq6wp7V8V0kx6naKxeTQPvEwRB5W3T4PxIKkk6VTUj4RH93w5eRLKfKJhAKIP9+9+G/8XBn4jlVLgF8Ir65+pDX8rHBHSnXqfIIoVONui/01MVX8DLbqbIIoVM9pwed2VkR+ANrs1BVTNFwtgAAAABJRU5ErkJggg==)
= ![](data:image/png;base64,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)
⇒ BC = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE4AAAA7CAMAAADim0xdAAAAAXNSR0IArs4c6QAAAGNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjqQOmaQOpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkGY6kNv/tmYAttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bt0JXFgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABjElEQVRYR+2Y63KCMBCFs7VVW5texMaiXPL+T9lsLiRBYoGsMzhjfjjjQD72kj2HgbFlr5YDwPrcBSnU36ef2TG33LDkAeDtqDEiHyeL9bkpVr9EuAojqmBPhBMYWMs3NDjMFXG6jvm1Mzjzu1QcWe2YbcU7TStYiWeE7qBgU20nCFrBWPMBsDOjm39QopF/4EYqoNO74Pb7qB3qfiD7V9MtlSegWF5JtuV65kauentT3OnTGphrhY1OntTAvCTNzW2Lo1NDZozHD5nFCfhmDXcXe6n7bRGu3h3LFA4NRCvO5Qq29WuXwmmI1a4Bots2CafldXDNwZlHU+FkkT6B06OThbZzqujExr9hXTAnR1fi4FapdJM49x7Wm4r6VU2EFEmc3RYdFHnYKk14/hqcCnUFYBAXbLuxBHRF9smS6t1IoYtuu1uveCT7TwVsZ0N/IXgLCP2FAhf4CwEu9BcqnPUXIpzzFxpc5y8kOO8vJDjvLxS4wF/ycPrTQucvmZ8W7AwiJeUvc4RqiXv+AO67M9Z2w4rjAAAAAElFTkSuQmCC)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ A(BC) = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD0AAAAmCAMAAABArWODAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGY6kLaQkLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29v/2////7Zm/9uQ/9u2//+2///bNexBegAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABYUlEQVRIS+2VUVPDIAyASacOda7OqVWHrqulg///Bw0UVtol6HzwPE8edtstH5AEPoT4NWN3CVDc57dTF5shwJQIhN918SDEDpY5vJMuWgH4MFPOY7Ct/Fd10fK4Kdf9WqZk6MN0xCRqqRlaaHA7T/Oa8vqqZWnRyKEI1O7N7UawdCcxlxoeubRthf8jXWMEkberl63YqmmstB8UHcqvZttcy7idh1WzHfM7JzuGKd+0n50WEc7aUd7YLTyp5y/Zk4q5+8UJ+oQL90+fUKxD6N+pWnMXdWdfJXPPBm2O896vIN4tLRfvdCETbY7o7vqtDrSW6CdypNqc1jzQiWSnU0TaaZOhNW+mVJsMrWbPK4AF7fRBmzRtK6eIxvnxeCTa5GjfLFJPvbr6zywdGzBaP9VmpOMDEIBeqZm13X/xFRwmD7rr5NypkXwSOG3aJ3yE4My93R2WnFPjF7T5nev+48wHwZkoWg+0H9AAAAAASUVORK5CYII=)
⇒ (AB) C = A (BC)
Thus verified.
Question 9.If
verify that (AB)T = BT AT.
Answer:⇒ AB = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIQAAAAmCAMAAAAyTsNwAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOmaQOma2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///b1lEYTwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACAklEQVRYR+1Ya1ODMBAk9VHUKq2KtakoQpr//xPN41IguUtDH6Mzkk9MuVs2d5tk0yybxt+twOfmGG7y656xm3Gp9aofvysYm22yNmdqqAdkyHppP8JVyJMJgCwbzGfvmSjxXHxSYsmuPnQmAO6KuQ5s81eyCJy9ZKIwWbvCkTBZQEI/V4wG8JHbxbYycA4whUT3EZSELQegJvZzNAmD25iZUiTEerFN/LwNO46EnSlOQitk8X0OEg95TOGgGbIddR5th1agHaBftBLyTYmPVrgsnSARYUK77JvEgZIwuW1OAMkSFgMtTDIXZ0WTgJUSpvE5dBwjYatkhZs8UBJcA1GzqW4Vh0ZH4CRUn0WpY9JHBdqwgDB7boBwcbV3areUhibaDrFW2+3jCA5SJ7Dr5z0gkBAr9TOx1kHaJIn0+YeRg0okAtHCTATwwyYSriJTJaZK/JfVMbSv6NbhhQxWR2VPe/QA6Lnp6Gbl7Gtk3wpChmeHPgMJt9tz0zESPftK0QhDwn3Cna8eBk8xuiaHAOjjeSEBCTdNdB4xj7lPOAMJd8YjJJybPnCAnU5i7+ECEp2b9kj4/vV0ElGDBm764pXgUYPWmFvoxUjAXeCAXdaviQtx176x7egAAYO8TI5w0xFpO6bREHp9Sn0nSnDTPftKbVaHQhr6/4XRbvpIv/mraT9GVzxz5vOsUQAAAABJRU5ErkJggg==)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ (AB)T = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE4AAAAmCAMAAABd010YAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAttv/tv//25A627Zm27aQ2////7Zm/9uQ/9u2//+2///bmT1U3gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABm0lEQVRIS+1W21KDMBBNqrWo1dKqlbapgYY0//+HZjchoCW7jDM+dKb7AAycnL1kswchrsaatwPE6pqNlHN8zFlEivpJytnHEHUu/RtYazfy7gu+KLkVtgzPo5aQerYVopYrWCPhJsS5XOCadnnUkQ5eaPmZY0tIV+FS9XBCnh90QDGIyOTpErKjQ1KaThHJJsfGl0XUWCqari2yucLaLo+miIWn6VyFhchapGsLD4tVJpKNReHoXAWbEK5UsmoBADa62BShzPnoNLgzVLoh2RhXtlHCHrWP/uYUSReQWr6eQhtfROd2fpfkvT8x0OFdk48l3CPxkM33CLpMlq4X8/VG9/fy3Wr3L7Vz2O2k+PR+dW58Joia7YWtAoyztuDpGPEZuDiX7ywdwmm16BjVykyhs7vlkcvTfzfPpwl0oOVLZiLjVFofxChdHHJ+zsUdaIpxaRwCUaE8nfayxx0yE6YsZSYMWgkSz9Gh6k2w38mm/Lq1QWXp34reD9KlX54R/w462Ab55C2eCgJoQYZeprH5+k07PnxgV4b4Bks4Kg9EKh9oAAAAAElFTkSuQmCC)
⇒ BT = ![](data:image/png;base64,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)
⇒ AT = ![](data:image/png;base64,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)
⇒ BT AT = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ (AB)T = BT AT
This proved.
Question 10.Prove that
are inverses to each other under matrix multiplication.
Answer:⇒ AB = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ BA = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIQAAAAmCAMAAAAyTsNwAAAAAXNSR0IArs4c6QAAAG9QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOmaQOma2OpDbZgAAZjoAZjo6ZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///b/YalIgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACHElEQVRYR+2Y2VLDMAxF47IEKIQlFAyBkqT+/2/Ea+JNssyUGYapn7rI6pEsKddtmtP6uxkQn9eMnb+SAcX+Hrf/oPk6dIxtnCnfvDRzv7wtwnD22Mzd2XvDGWN3xtw6nFr50eo5cLWwu22H7nI14Or1wJ6Kv24NPPtDt0Boh1MLe1nYFbLaFkBo31xGVrNGDV0Fscaah5h327caBAddA6H9++xhJtSBbr+qIGzeqyFMwoHj2LfocahaMsvUr+hNLaQQNy3WaAF7WhOjq3NKQkRv6zqGEM+yceBGC9lTiKm1dU6B4Jf28GIIvRl0FbH7EAbPVAxtDReSYTRdFraoPSOv/32PEXsIIc957pVj2pquZF0InoPQH0KZiNmD45h3cs7dkhn0oLSzMs4E1/HkazxhT2uCloTYKoaYHyQdMHIS9t+CIIYCjW3i9sgsW5hlVycIl6NTJv5bJjwJefTuGMwgxMZ3IiGPDsHVEwwTiZlvYwgsEE/UF7pjQLVNgpg8O5BAPFGPQyxO81OvCKG3AYEkIh16dgz47YMEgQaCakwT+SJ9gPE/tZGEzBUmEogT9cFxxPq1JK8SCZmBgANZRT1aE5wir3zhlIFAA7GiHoOgiV2/njIQeCBG1DuI3LW1eBWMJGR6IUaktikzFae3La29Qn/KDZiEtP7AQIiifiz+O4BJSAMBByLUnahG1JcFGmSBBFIr6n8OQd/5DWoKOXfK1T2hAAAAAElFTkSuQmCC)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Thus A and B are inverse to each other
Question 11.Solve
.
Answer:Let A = [x 1] B =
C = ![](data:image/png;base64,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)
⇒ BC = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ A(BC) = [x 1]![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
=
= 0
∴ x = -3, x = 5
Question 12.If
, find (A + B)C and AC + BCIs (A + B)C = AC + BC ?
Answer:⇒ A =
, B =
and C = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEEAAAAmCAMAAACs2AaVAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZmZmZrbbZrb/kDoAkGY6kLaQkLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29v/2////7Zm/9uQ/9u2//+2///b/GlvyAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABYUlEQVRIS+2W3VqDMAyGm03X6eaqm8xRlW307/7v0DZFKFBadqY+5gAOSF+SLyGBkB9k5vICcFfmIzLnh4GjZgCLknA4EMWWn1kEX5yIKvAEwM65a7ZyN+6uFbzmCZ2jZiEBT4oZBHTkGGyE4B9kTR03Hz78YQyS5pNw5wA2dZRgCkSOzWnmzQro7EKjWZgCJZ1lAqswzIKvMLRZJmmEUN1bgJjIo8P6VH3R+jHItU3R8BkEq4Uq3OsGhEatLIGoIwXYxmsxS4HQKdJRNzL+CV6wv6pD1Xy8k21h3im2c2MjHSTNEATdXlM9qdk+TRD00I9uNB92IklohnoAGRDEY50mjAd5n6CfS5Im8OWbXWv+u450FI4eS6gmp7Up4Km2MzaYIL0YxPc0ThCwkDwo57irk1kYP9mqYCm1hHYN5HRwC62Lod28QXXSPSmpXQfn5Ga1WiQbQro/jNONk/CXuH8BeuwhIoaDqFkAAAAASUVORK5CYII=)
⇒ A + B = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ A + B = ![](data:image/png;base64,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)
⇒ (A + B)C = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEcAAAAmCAMAAAChxnbSAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOjqQOma2OpC2OpDbZgAAZgA6ZjoAZmZmZrbbZrb/kDoAkGY6kLaQkLbbkNv/tmYAtmY6tpA6ttv/tv//25A627Zm27aQ29v/2////7Zm/9uQ/9u2//+2///b5P9XEAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB0klEQVRIS+1W21YCMQxsRamKoMIK1EtFutL//0ObpPdN5dWj9mE5bNJpkkkzK8TPXe5ZXR2FcIcHKS/3bJzZZiQu2EDrtJLyAnZZtfiAF1quxedq9sYBZZveePuoNt5dyiXizHGHVWvaqeG/keA3WY3N4GmnVYET4cJWy+PQidFGADVOs1HzeVHA0WawHjWOnm19eRehbJB4byWbG6giZV5ukHdHcVAYqHAD/XIr22IKNQ4yqPEZD+JhKIjk3MaDCESAnqeumEBl2xiCb+uDCIBm4GH5zApbKnfF+6h8EO/A6HjtWXCaxSlsgfSWLzHCbdhBTNTxLE5hs0T6BKdL0FlDlddZ777DP873xfsT9anndphQ7IUnNRCoBq+Tfk5zGyyjSk3fQkU1MBc7fwmn8xn9aWz4QfDYw4lqEDmiCV/qRXrhb+oyX8I6nqQGNFzdAClOcEJV7O2xh9NIBY8Txunpfi96OFkN9Gwv3AvqaRsPHYZz3OMYRjEKNXBPnq4tamiLQ0Pe0hiTnBBSHkENkFemzmluAxjPV8AJtHp+0S3GE74TSgnt1yepAQL4tOJ3QiQ2z21opE7/ZDWAjlZRfsvmKEPwNeoBRTXwtbkBVfjl6wsFAy/MODY6fQAAAABJRU5ErkJggg==)
⇒ AC = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ BC = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ AC + BC = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ Thus (A + B)C = AC + BC is true
Determine whether the product of the matrices is defined in each case. If so, state the
order of the product.
(i) AB, where A = [aij]4x3, B = [bij]3x2
(ii)PQ, where P = [pij]4x3, Q = [qij]4x3
(iii)MN, where M = [mij]3x1, N = [nij]1x5
(iv) RS, where R = [rij]2x2, S = [sij]2x2
Answer:
(i) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.
⇒ Here A[aij]4 x 3 and B = [bij]3x2
⇒ Number of columns in A = 3
⇒ Number of rows in B = 3
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 4 × 3
(ii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.
⇒ Here P[pij]4 x 3 and Q = [qij]4x3
⇒ Number of columns in P = 3
⇒ Number of rows in Q = 4
Thus the product is not defined.
(iii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.
⇒ Here M[mij]3 x 1 and N = [nij]1x5
⇒ Number of columns in M = 1
⇒ Number of rows in N = 1
Thus the product is defined and the order if product is
Number of rows in M × Number of columns in N
∴ MN = 3 × 5
(iv) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.
⇒ Here R[rij]2 x 2 and S = [sij]2x2
⇒ Number of columns in R = 2
⇒ Number of rows in S = 2
Thus the product is defined and the order if product is
Number of rows in R × Number of columns in S
∴ RS = 2 × 2
Question 2.
Find the product of the matrices, if exists,
(i) (ii)
(iii) (iv)
Answer:
(i) ⇒ let A : [2 -1] ∴ A[aij]1 × 2
⇒ let B : ∴ B[bij]2 × 1
Number of columns in A = 2
Number of rows in B = 2
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 1 × 1
⇒ [2 × 5 + (-1) × 4]
⇒ [10-4]
⇒ [6]
(ii) ⇒ let A : ∴ A[aij]2 × 2
⇒ let B : ∴ B[bij]2 × 2
Number of columns in A = 2
Number of rows in B = 2
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 2 × 2
⇒
⇒
(iii) ⇒ let A : ∴ A[aij]2 × 3
⇒ let B : ∴ B[bij]3 × 2
Number of columns in A = 3
Number of rows in B = 3
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 2 × 2
⇒
⇒
(iv) let A : ∴ A[aij]2 × 1
let B : [2 7] ∴ B[bij]1 × 2
Number of columns in A = 1
Number of rows in B = 1
Thus the product is defined and the order if product is
Number of rows in A × Number of columns in B
∴ AB = 2 × 2
× [2 -7]
⇒
⇒
Question 3.
A fruit vendor sells fruits from his shop. Selling prices of Apple, Mango and Orange are ₹20, ₹10 and ₹5 each respectively. The sales in three days are given below
Write the matrix indicating the total amount collected on each day and hence find the total amount collected from selling of all three fruits combined.
Answer:
⇒ Let the Sales matrix be A =
⇒ Selling price matrix B =
AB =
=
=
These are the amounts earned on each day.
Total amount earned = 1750 + 1600 + 1650 = 5000 Rs
Question 4.
Find the values of x and y if .
Answer:
=
=
=
Comparing with
3x = 9
∴ x = 3
And Y = 0
Question 5.
If and if AX = C, then find the values of x and y.
Answer:
x = 2, y = - 5
⇒ AX =
=
Comparing with
5x + 3y = -5 ---1
7x + 5y = -11 ---2
Multiply 1 by 5 and multiply 2 by 3 and subtract,
x = 2 and y = -5
Question 6.
If then show that A2 = 4A + 5I2 = O.
Answer:
⇒ A2 = AA =
=
=
⇒ 4A =
=
⇒ 5I =
⇒ A2 - 4A + 5I2 =
=
=
Question 7.
If then find AB and BA. Are they equal?
Answer:
⇒ AB =
=
=
⇒ BA =
=
=
AB not equal to BA
Question 8.
If verify (AB) C = A (BC).
Answer:
⇒ AB =
=
=
⇒ (AB)C =
=
=
⇒ BC =
=
=
⇒ A(BC) =
=
=
⇒ (AB) C = A (BC)
Thus verified.
Question 9.
If verify that (AB)T = BT AT.
Answer:
⇒ AB =
=
=
⇒ (AB)T =
⇒ BT =
⇒ AT =
⇒ BT AT =
=
=
⇒ (AB)T = BT AT
This proved.
Question 10.
Prove that are inverses to each other under matrix multiplication.
Answer:
⇒ AB =
=
=
⇒ BA =
=
=
Thus A and B are inverse to each other
Question 11.
Solve .
Answer:
Let A = [x 1] B = C =
⇒ BC =
=
=
⇒ A(BC) = [x 1]
=
= = 0
∴ x = -3, x = 5
Question 12.
If , find (A + B)C and AC + BCIs (A + B)C = AC + BC ?
Answer:
⇒ A = , B =
and C =
⇒ A + B =
=
⇒ A + B =
⇒ (A + B)C =
=
=
⇒ AC =
=
=
⇒ BC =
=
=
⇒ AC + BC =
=
⇒ Thus (A + B)C = AC + BC is true
Exercise 4.4
Question 1.Which one of the following statements is not true?
A. A scalar matrix is a square matrix
B. A diagonal matrix is a square matrix
C. A scalar matrix is a diagonal matrix
D. A diagonal matrix is a scalar matrix.
Answer:In the above question we see the following terms,
Scalar Matrix, Square Matrix, Diagonal Matrix
To answer this question, we have to know the definition of the above terms.
Square Matrix:
If the rows and columns of the matrices are equal then it will constitute square like structure. So, it is called as square matrix.
E.g.![](data:image/png;base64,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)
Diagonal Matrix:
In a square matrix all the elements are zero except the diagonal elements of the matrix. Then that matrix is said to be diagonal matrix.
E.g.![](data:image/png;base64,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)
Scalar Matrix:
In a diagonal matrix, if all the diagonal elements are same then that matrix is said to be a scalar matrix.
E.g.![](data:image/png;base64,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)
Option(A):
A scalar matrix should be a square matrix so, it is true
Option(B):
Diagonal matrix forms only with the square matrix so, it is true
Option(C):
A scalar matrix consists of zero except the diagonal elements so it is true.
Option(D):
A scalar matrix should comprise of same diagonal elements but in diagonal matrix it may (or) may not contains same diagonal elements. So it is False.
Option(D) is not True in the given.
Question 2.Matrix A-[aij]mxn is a square matrix if
A. m < n
B. m > n
C. m = 1
D. m = n
Answer:Given that A is a matrix with m and n as their rows and columns respectively.
We know that for a square matrix both m and n should be equal
So, m = n is the correct answer.
Question 3.If
then the values of x and y respectively are
A. –2, 7
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. 2, –7
Answer:Given,
= ![](data:image/png;base64,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)
If the one matrix is equivalent to other matrix, then their elements should be equal.
⇒ 3x + 7 = 1 & 5 = y-2
3x = -6 ; y = 7
x = -2 ; y = 7
∴ Option (A) is the answer.
Question 4.If A = (1 -2 3) and
then A + B
A. (0 0 0)
B. ![](data:image/png;base64,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)
C. (-14)
D. not defined
Answer:For matrix addition operation their no. of rows and no. of columns should be equal. Otherwise addition is not possible.
In the given matrix A has 1 row and 3 columns
But matrix B has 3 rows and 1 column.
Since rows and columns are not equal it is not possible to add.
So the answer is (d)
Question 5.If a matrix is of order 2 × 3, then the number of elements in the matrix is
A. 5
B. 6
C. 2
D. 3
Answer:Given that a matrix with order 2x3
We know that the no. of elements in the matrix is equal to the product of no. of rows and no. of columns
i.e., No. of elements = No. of rows × No. of columns
∴ No. of elements = 2 x 3
= 6
∴ There will be 6 elements in the given matrix
So, option (B) is the correct answer.
Question 6.If
then the value of x is
A. 1
B. 2
C. ![](data:image/png;base64,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)
D. 4
Answer:Given,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Since both the matrix is equal to each other, the elements in it also be equal.
By equalizing we will get x = 4
So, option (D) is correct answer.
Question 7.If A is of order 3 × 4 and B is of order 4 × 3, then the order of BA is
A. 3 × 3
B. 4 × 4
C. 4 × 3
D. not defined
Answer:Given matrix orders
[A] with 3 x 4 and [B] with 4 x 3
We have to find the order of [B].[A]
For matrix multiplication the columns of 1st matrix and the rows of 2nd matrix should be equal.
E.g. [A]3x2.[B]2x2
The order of new matrix formed after multiplication of matrices will be 1st matrix row will be taken as new matrix rows. The no. of columns in 2nd matrix will be taken as new matrix columns.
E.g. [A]3x2.[B]2x2 = [AB]3x2
In the same way [B]4x3.[A]3x4 = [AB]4x4
∴ option (B) is correct answer
Question 8.If
, then the order of A is
A. 2 × 1
B. 2 × 2
C. 1 × 2
D. 3 × 2
Answer:Given,
A x ![](data:image/png;base64,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)
Let us consider
as matrix B, it has an order of 2x2
Let us consider the resultant matrix as C, it has an order of 1x2
For matrix multiplication the columns of 1st matrix and the rows of 2nd matrix should be equal.
So, matrix A has got same Columns as Rows of matrix B
The order of new matrix formed after multiplication of matrices will be 1st matrix row will be taken as new matrix rows. The no. of columns in 2nd matrix will be taken as new matrix columns
So, matrix A has got same Rows as Rows of matrix C
So, Matrix A will consists of 1x2 order
∴ option (C) is correct answer
Question 9.If A and B are square matrices such that AB = I and BA = I, then B is
A. Unit matrix
B. Null matrix
C. Multiplicative inverse matrix of A
D. –A
Answer:It is based on a property about multiplication inverse. Only product of a matrix and its own inverse matrix then the resultant matrix will be identity matrix.
AB = I and BA = I
So, B is the multiplicative inverse matrix of A
∴ option (C) is correct answer.
Question 10.If
, then the values of x and y respectively, are
A. 2, 0
B. 0, 2
C. 0, -2
D. 1, 1
Answer:Given,
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
By equating all the elements in the matrix
We will get 2 equations
x + 2y = 2 ⇒ 1
2x + y = 4 ⇒ 2
By solving the both equations we will get
x = 2; y = 0
∴ option (A) is correct answer
Question 11.If A =
and A + B = O, then B is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Given that
A + B = O ⇒ B = -A
∴ B = ![](data:image/png;base64,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)
B = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFIAAAAmCAMAAABgQTx+AAAAAXNSR0IArs4c6QAAAIdQTFRFAAAA///b//+22///tv///9u2/9uQttv/kNv//7Zm27a227aQtrbb27ZmkLbbZrb/Zrbb25A6ZpDbOpDbkGZmtmYAkGY6ZmZmOma2OmaQkDo6AGa2kDoAZjo6OjqQZjoAOjpmOjo6OjoAADqQADpmADo6ZgA6ZgAAOgA6OgAAAABmAAA6AAAA8Qs0bwAAAAF0Uk5TAEDm2GYAAAGKSURBVHja1VZdV4MwDG3p+KioiLUTBjoVFZH8/9/noaNAC5QVtwfywDmU5vYmTXJBaOMWvgTG7zg8AnzO7nGffH3lCL87IySHB+SW/SaakMFX710/zEv9eAkyQwjFEHUL8Vvv4JTRhMsS5InaABLFBVHOWwfJh5twzmSQP8FaSK9S4qPSpSXJQVodnAnZ01LftZMsWOI8m/aZc12G5BmZTgQvyDpIccFUCd0pmXhmMx61uXvErWLO1FQ0x3gVm0rTvgKAr0dj9whj46JSivXfJkK7AuRSyiwhG4KXhaRXgtzG9cy1+DoTdTld6qshm+7Rxwm++zCplLl3WzBtbPD6Hrm5VRl4ldzehkxV95FKLZpTPkuIFus0kAySspg/1rGSEcf6wHT3qW9R3a9EQnZSq2muUwKk5Pywv4Mudz25Ec2wMgWu6J5QMFoHcaRyGwk5BXZ2U7cWqXqJDwnRysKq/kXg+JAoizc7RYotG19Aurez2lwHyM0Lsqp7Zv4LGyFLiN2QhIsO3Y3aHwE7JZqQPpVHAAAAAElFTkSuQmCC)
∴ option (B) is correct answer
Question 12.If A =
, then A2 is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Given A = ![](data:image/png;base64,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)
A2 = A×A =
x ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
∴ option (D) is correct answer
Question 13.A is of order m x n and B is of order p x q, addition of A and B is possible only if
A. m = p
B. n = q
C. n = p
D. m = p, n = q
Answer:Given matrix of A is m x n ⇒ [A]mxn
Matrix of B is p x q ⇒ [B]pxq
We know that addition is possible is possible only the order is same for both the matrices
So, m = p, n = q
So, option (D) is true
Question 14.If
, then the value of a is
A. 8
B. 4
C. 2
D. 11
Answer:
= ![](data:image/png;base64,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)
By multiplying the matrices, we will get the value of ‘a’
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
By equating the elements in the matrices
2a-3 = 5
2a = 8
a = 4
So, option (B) is correct answer
Question 15.If A =
is such that A2 = I, then
A. 1 + α2 + βγ = 0
B. 1 – α2 + βγ = 0
C. 1 – α2 – βγ = 0
D. 1 + α2 – βγ = 0
Answer:Given A = ![](data:image/png;base64,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)
A2 = I
I is the identity matrix
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
![](data:image/png;base64,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)
By comparing the elements in the matrix
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIQAAAAXCAMAAAD9aBd4AAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjqQOmZmOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkGY6kLaQkLbbkLb/kNu2kNv/tmYAtmY6tmZmtpBmtrZmttv/tv//25A627Zm27aQ27a229u229v/2////7Zm/9uQ/9u2//+2///b80zxNwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABpUlEQVRIS+1V21LCMBDNtoJUFCgVKV5oVUxRm5L8/8+5SQoWai4w1dEZ9yHDMLtnz569lJB/O1qBtwRgfHRUtwF8ekeKYNkt6Clo1YWZhHiKIJidgqpj1teeBdKJOUkWrkgBFod2qFgAXK7U/5sEwhevCgrLSPAY84t04AWknUTaLzepyl2NVtSPBLWNJZPjchwJFcJgrilZSVQJ9JayToYcWB3RrpdKRHQTOYxK+auPr90yWTuPa/FsJKpoRorwOUAegGYmgYibZFzyeyYHY1egmYfsRoOtjUSGniIdOnuNGBTO32VOGZI1AzJJX1lzxTUJ/e6349BfzRuhLQVauLIQkQdSKew1n7r3zUziUD1NgrkbrNWUIpAqmjOnclsNfGZCk1AjZDdNQr0ivcn3hPi6HRpVJ7Bvh1o6nrjPtdqOmm125fWNUU3+XFFLDgQXjw/x5NWhBYVBSdaRKksTcppc43ouxSLCsT0zXX08rb0lyWHsWHsa3g633w6PEVIM8Varo9KZNdecuue4s7x7QA0Su9X/nkwW1N9GgsfG4/7j2vyxhB/mLCcAjvfoiQAAAABJRU5ErkJggg==)
∴ Option (C) is correct answer.
Question 16.If A = [aij]2x2 and aij = i + j, then A =
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADcAAABACAYAAAC+/O8/AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAU1SURBVGje7ZtPTxtHFMCfHYlToMcoCeNT05Bb2DHfgJhxIJE4rGC9siKl0h5WaNVgpD3QsHCzCuk5Ug4NuRR86K1SFdJ7Q75BAx8gJR8hrTtv1oZlPfbO/iHYyR6eZNDueH4z79/Me4atrS34UuWLBcvhxEMABZShmHCMuUQ+0AIoOia7z5j52Go2r14mWLNpXTUZe8xM5z5Aq5gKzrX0CZOSnRnTWRomdXPMmSVCzR3dcicSwSFYjZJntL45P4z2tFmn84TWngG0C7HgUKcRjDB7VcEGilyudKSYxjbDY0U9bzOySiora/2+U/qSZ9AHhFT3BtkYDuja+m2NwDv+uS2EaO8qS/ZCXEB8ft3Wv6ME/uKfPwkh2qFuu7ejbLBKyB41vAdKcK5rjc8Rss8am3cHDbxuz/K5FE5OJ8OFu7H/EHLaePowDhwfSyuX2U/dxfQc45YPqh3XXffGQPVssLuEzO1brjseCYe7BtR8GbVijBQOCDM2ghOaI/DaB6THdc+7ruoBGWU/hrXEc9g0AfhIzc2FqDFMCi9lu9f7RZPwJsqJ4GrNsJXvpRPl7wOQj8zxptM4DNxN1AwVh4bOBSbZm/ACQe9qaYeqqy6TGoVdvnNHScfoBOmiQYu7QGu/qLzjefXrGpDD8ILGVslIFcOdU5yULGGwbZ0KJ0Urv0XFsSjV7Fl1FR0frEr0bZJdw4VBm+04p3/RMRFt+WfLbY4rxT2TLvBF3ZXC+U6Ce8mEttLuxMa0QR/nITIQgH8EII9jSqqJJkXYftDuQnqrHSRZdbSTlQppqAR95QwEXTzAiarnlc0/NRyCoa3KvGdaQTvKBk6yrWqZxaxGqbEhS6XSwnFtWAPCDlTmJDOrxHAI9tSYfohGH8gHhTSsxSlK2bbqWLokL8XE3U8U1FQ9MzhhYwatYQZxmleGhDBnVc0R8FQL4AjzUt22KS6O1Tlq8ax/O1YYygIOvVkwp+yV0oeo3DQ4qUq59Dz4fqlU/p2ZT6qxY2xWajlsogLXGnG4lhzu0dTi2Nid99yVXhtFOB4Krt0ZG3s/9chb/CrVMofL4XK4HC6Hy+FyuBwuh8vhcriLPs+1xVW4fiXNJPXQfUxHClmc546Snuc6d5drcSo8YXHd+g0N4Pj8fQw5Ubmy6JznjuTnuZRwfr0OL4ySw+HFbs99jHox5GLgcMUpKbyNW5s7NzlXv0lL917oljWRpPR8IXBn9XNzw6/NxYcTY2jwSlwJEvpHp/RczBou9h2KqJ9Tc+es8BgfDusCJYAPwQqPuMd03ZuXdoeC6lgm5V8RJg1cV2zb/tZkMz8gqK/i2rHuekqAmca5bsmq68mygAtOtOrX19ug1V6lh4tR5ZGVrLKE685HXLN/7hKWqKCGCiCWpX9TmYQ/hSo5zmTaYI7iMLKq2jwwEC5OZdUWX9pbI/DtpND2/9b+TruDcRoQxIL3q6yi9OvnUNb5DNWy64mBVF+r+AGRHYWaFc67ZEnR/HPAtURt7nyLIf4Py1iq1SJZs0KvGvRpNbpIOIQo0XsvFq3GVLA+p9oK6eekEX0ovt3BQRLVPGu1iG9rWO8LBPFPpXLlueF4t+Kpb295uTdbqNP5IlfNYWn7VUv/iruyFpE+AZTszdrr2ijA+V5S3j4pz/WwCxXdqmL3zqWdvl1rHMNXv8aeAd6HbJPKSmNY1fM0SxrQFDDQQSAgOpdhA/S7KWYMBBvYzRvlARGQu+TlYfpdAfeuy1FgkXBdeWKy6nD9rkCtjWPkrvBi7XION6LyPw8w3OJa083LAAAAAElFTkSuQmCC)
D. ![](data:image/png;base64,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)
Answer:Given A = [aij]2x2 and aij = i + j
It means matrix A with order 2x2 and the elements in is given by aij = i + j
a11 = 1 + 1 = 2
a12 = 1 + 2 = 3
a21 = 2 + 1 = 3
a22 = 2 + 2 = 4
A =
= ![](data:image/png;base64,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)
∴ Option (B) is correct answer.
Question 17.
, then the values of a, b, c and d
respectively are
A. -1, 0, 0, - 1
B. 1, 0, 0, 1
C. -1,0,1,0
D. 1, 0, 0, 0
Answer:Given
![](data:image/png;base64,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)
By multiplying the matrices, we will get the values of a, b, c & d
![](data:image/png;base64,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)
![](data:image/png;base64,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)
By comparing the matrices on both sides
-a = 1; -b = 0; c = 0; d = -1
∴ Option (A) is correct answer.
Question 18.If
, then the matrix B =
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Given
A + B =
; A = ![](data:image/png;base64,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)
B = ![](data:image/png;base64,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)
B = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKAAAAAmCAMAAAB0+GDaAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLbbkNv/tmYAtrbbttv/tv//25A627Zm27aQ27a22////7Zm/9uQ/9u2//+2///bsnbhwgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC6klEQVRYR+1Ya1PUMBRtQJf6YKWCSF0lYl23TfP/fx953ySbe5t2BkdnyAeGock5575yc2ma1/ViHjjePdLY8sDY9dNq/oGZtTsVTsrjLWNvC7zie04kbtnlL5Jc9ruT6GHT+KlEmUJoHv6g/ja1+ufZ4uxrIzqFmYFNH3LR0/5pWBA4XqhDIwOi4SNtkZIVeBBwfqU0DxozAZu7kjlLArk2YO40pFtDMWzwHXjm7gaPjjU6BjO6z9aCQB1hLTASJXuCVuEDz6C9jy1jeROBTe+Ku6sEWpl+jWRWAI/siy6xMD49AcwZxm11qWXN2yCQdiE4MM7c3I8BI/yCFFSdwCQHaaMiHo4na+Rc7yHMU5VFkqQdZmwakanFc5VfhZTxYJg5SwLNbZAFi6pO4LFVUFymdker34FlQYJjZKWZ87tTWiO69NDgAQ9hhakjya1AB1b2tzy0qlre3OOXQdOoZsP2WffAnQM85oYvL1eoLgMsGFVRlDwkRCj5Bh4bww0HCd14XmzgsWBLqbbOi6ZyyulPNQ/kiO156w8SmnE/beBxTflFBOZtaZMjrMANufF3c/CfF0h1p3UFondT9yBWPiiLBaMa4waBaCfZwGObY/42Q+eWmswjHnrxJ3Qek7/fw9DkT2SPhTC31Ltvav1NQL5TvHOJeYxf/GhE79A8WNYaYW6pVTh3X7xAos02/hs1jyXk/kDpcbGqtPlN0EW8RP3rSZtNv+R8oQWwwjBGvNjO/Dpen8BWqlKBhxIoDns7o8czYA675uqZPz+G6NFzJzDiAucuPOEisBx3YXxM2pjZqzw4KCPL8zW4PPCQIT625vpLwNLBmBwK8wiPfhR8iCdZpL48D52DI1M2p06S3+L/q0RzS2UhmxxUIEvbPQ8tUF8vZ2B/YIyJ55YlRvfdCBQ/K3ZbHkygdZu+QgiwZG6poDSEK59s2H6p72iBD15GTjq3VClUebhCITWPCT2rVfwbr0rW66b/wwPP4AFK+tWPh8QAAAAASUVORK5CYII=)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
∴ Option (C) is correct answer.
Question 19.If
, then the value of x is
A. 7
B. -7
C. ![](data:image/png;base64,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)
D. 0
Answer:Given ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
By equating the elements
13 – x = 20
x = -7
∴ Option (B) is correct answer.
Question 20.Which one of the following is true for any two square matrices A and B of same order?.
A. (AB)T = ATBT
B. (ATB)T = ATBT
C. (AB)T = BA
D. (AB)T = BTAT
Answer:This is Reversal law for transpose of matrices
∴ (AB)T = BTAT
∴ Option (D) is correct answer.
Which one of the following statements is not true?
A. A scalar matrix is a square matrix
B. A diagonal matrix is a square matrix
C. A scalar matrix is a diagonal matrix
D. A diagonal matrix is a scalar matrix.
Answer:
In the above question we see the following terms,
Scalar Matrix, Square Matrix, Diagonal Matrix
To answer this question, we have to know the definition of the above terms.
Square Matrix:
If the rows and columns of the matrices are equal then it will constitute square like structure. So, it is called as square matrix.
E.g.
Diagonal Matrix:
In a square matrix all the elements are zero except the diagonal elements of the matrix. Then that matrix is said to be diagonal matrix.
E.g.
Scalar Matrix:
In a diagonal matrix, if all the diagonal elements are same then that matrix is said to be a scalar matrix.
E.g.
Option(A):
A scalar matrix should be a square matrix so, it is true
Option(B):
Diagonal matrix forms only with the square matrix so, it is true
Option(C):
A scalar matrix consists of zero except the diagonal elements so it is true.
Option(D):
A scalar matrix should comprise of same diagonal elements but in diagonal matrix it may (or) may not contains same diagonal elements. So it is False.
Option(D) is not True in the given.
Question 2.
Matrix A-[aij]mxn is a square matrix if
A. m < n
B. m > n
C. m = 1
D. m = n
Answer:
Given that A is a matrix with m and n as their rows and columns respectively.
We know that for a square matrix both m and n should be equal
So, m = n is the correct answer.
Question 3.
If then the values of x and y respectively are
A. –2, 7
B.
C.
D. 2, –7
Answer:
Given,
=
If the one matrix is equivalent to other matrix, then their elements should be equal.
⇒ 3x + 7 = 1 & 5 = y-2
3x = -6 ; y = 7
x = -2 ; y = 7
∴ Option (A) is the answer.
Question 4.
If A = (1 -2 3) and then A + B
A. (0 0 0)
B.
C. (-14)
D. not defined
Answer:
For matrix addition operation their no. of rows and no. of columns should be equal. Otherwise addition is not possible.
In the given matrix A has 1 row and 3 columns
But matrix B has 3 rows and 1 column.
Since rows and columns are not equal it is not possible to add.
So the answer is (d)
Question 5.
If a matrix is of order 2 × 3, then the number of elements in the matrix is
A. 5
B. 6
C. 2
D. 3
Answer:
Given that a matrix with order 2x3
We know that the no. of elements in the matrix is equal to the product of no. of rows and no. of columns
i.e., No. of elements = No. of rows × No. of columns
∴ No. of elements = 2 x 3
= 6
∴ There will be 6 elements in the given matrix
So, option (B) is the correct answer.
Question 6.
If then the value of x is
A. 1
B. 2
C.
D. 4
Answer:
Given,
Since both the matrix is equal to each other, the elements in it also be equal.
By equalizing we will get x = 4
So, option (D) is correct answer.
Question 7.
If A is of order 3 × 4 and B is of order 4 × 3, then the order of BA is
A. 3 × 3
B. 4 × 4
C. 4 × 3
D. not defined
Answer:
Given matrix orders
[A] with 3 x 4 and [B] with 4 x 3
We have to find the order of [B].[A]
For matrix multiplication the columns of 1st matrix and the rows of 2nd matrix should be equal.
E.g. [A]3x2.[B]2x2
The order of new matrix formed after multiplication of matrices will be 1st matrix row will be taken as new matrix rows. The no. of columns in 2nd matrix will be taken as new matrix columns.
E.g. [A]3x2.[B]2x2 = [AB]3x2
In the same way [B]4x3.[A]3x4 = [AB]4x4
∴ option (B) is correct answer
Question 8.
If , then the order of A is
A. 2 × 1
B. 2 × 2
C. 1 × 2
D. 3 × 2
Answer:
Given,
A x
Let us consider as matrix B, it has an order of 2x2
Let us consider the resultant matrix as C, it has an order of 1x2
For matrix multiplication the columns of 1st matrix and the rows of 2nd matrix should be equal.
So, matrix A has got same Columns as Rows of matrix B
The order of new matrix formed after multiplication of matrices will be 1st matrix row will be taken as new matrix rows. The no. of columns in 2nd matrix will be taken as new matrix columns
So, matrix A has got same Rows as Rows of matrix C
So, Matrix A will consists of 1x2 order
∴ option (C) is correct answer
Question 9.
If A and B are square matrices such that AB = I and BA = I, then B is
A. Unit matrix
B. Null matrix
C. Multiplicative inverse matrix of A
D. –A
Answer:
It is based on a property about multiplication inverse. Only product of a matrix and its own inverse matrix then the resultant matrix will be identity matrix.
AB = I and BA = I
So, B is the multiplicative inverse matrix of A
∴ option (C) is correct answer.
Question 10.
If , then the values of x and y respectively, are
A. 2, 0
B. 0, 2
C. 0, -2
D. 1, 1
Answer:
Given, =
=
By equating all the elements in the matrix
We will get 2 equations
x + 2y = 2 ⇒ 1
2x + y = 4 ⇒ 2
By solving the both equations we will get
x = 2; y = 0
∴ option (A) is correct answer
Question 11.
If A = and A + B = O, then B is
A.
B.
C.
D.
Answer:
Given that
A + B = O ⇒ B = -A
∴ B =
B =
∴ option (B) is correct answer
Question 12.
If A = , then A2 is
A.
B.
C.
D.
Answer:
Given A =
A2 = A×A = x
=
=
=
∴ option (D) is correct answer
Question 13.
A is of order m x n and B is of order p x q, addition of A and B is possible only if
A. m = p
B. n = q
C. n = p
D. m = p, n = q
Answer:
Given matrix of A is m x n ⇒ [A]mxn
Matrix of B is p x q ⇒ [B]pxq
We know that addition is possible is possible only the order is same for both the matrices
So, m = p, n = q
So, option (D) is true
Question 14.
If , then the value of a is
A. 8
B. 4
C. 2
D. 11
Answer:
=
By multiplying the matrices, we will get the value of ‘a’
=
By equating the elements in the matrices
2a-3 = 5
2a = 8
a = 4
So, option (B) is correct answer
Question 15.
If A = is such that A2 = I, then
A. 1 + α2 + βγ = 0
B. 1 – α2 + βγ = 0
C. 1 – α2 – βγ = 0
D. 1 + α2 – βγ = 0
Answer:
Given A =
A2 = I
I is the identity matrix
=
By comparing the elements in the matrix
⇒
∴ Option (C) is correct answer.
Question 16.
If A = [aij]2x2 and aij = i + j, then A =
A.
B.
C.
D.
Answer:
Given A = [aij]2x2 and aij = i + j
It means matrix A with order 2x2 and the elements in is given by aij = i + j
a11 = 1 + 1 = 2
a12 = 1 + 2 = 3
a21 = 2 + 1 = 3
a22 = 2 + 2 = 4
A = =
∴ Option (B) is correct answer.
Question 17.
, then the values of a, b, c and d
respectively are
A. -1, 0, 0, - 1
B. 1, 0, 0, 1
C. -1,0,1,0
D. 1, 0, 0, 0
Answer:
Given
By multiplying the matrices, we will get the values of a, b, c & d
By comparing the matrices on both sides
-a = 1; -b = 0; c = 0; d = -1
∴ Option (A) is correct answer.
Question 18.
If , then the matrix B =
A.
B.
C.
D.
Answer:
Given
A + B = ; A =
B =
B =
=
=
∴ Option (C) is correct answer.
Question 19.
If , then the value of x is
A. 7
B. -7
C.
D. 0
Answer:
Given
By equating the elements
13 – x = 20
x = -7
∴ Option (B) is correct answer.
Question 20.
Which one of the following is true for any two square matrices A and B of same order?.
A. (AB)T = ATBT
B. (ATB)T = ATBT
C. (AB)T = BA
D. (AB)T = BTAT
Answer:
This is Reversal law for transpose of matrices
∴ (AB)T = BTAT
∴ Option (D) is correct answer.