xii matrices and determinants work sheet (1)

D D M S P. OBUL REDDY PUBLIC SCHOOL CLASS XII MATHEMATICS WORK SHEET Matrices and Determinants KEY POINTS: 1. If A is any non singular matrix then (A-1)T = (AT)-1 2. i) (A ± B)T = AT±BT ii) (AB)T = BT AT iii) (AB)-1 = B-1A-1 3. For any two square matrices A and B of same order, then I AB I = IAI IBI 4. If A is a square matrix of order n, then A (adjA) = (adjA) A = IAI. I 5. If A is invertible matrix of order n, then IadjAI = IAIn-1. 6. If A is a square matrix of order n, then IA.adjAI = IAIn 7. If A and B are non singular matrices of same order, I is unit matrix, k is non zero real number such that (i) AI = IA = A (ii) AB = BA =I then ������−1 = B , ������−1 = A (iii) AB = BA = k I then ������−1 = 1 B , ������−1 = 1 A ������ ������ 8. Det(������−1) = 1 ������������������ ������ Questions: 1. If A and B are square matrices of the same order such that IAI = 6 and AB = I, then write the value of IBI. (Ans: 1/6) 2. If A is a skew symmetric matrix of order 3, write the value of IAI. (Ans: 0) 3. If A = [−������������������������������������������������ ������������������������������������������������] , then for any natural number n, find the value of Det(An). (Ans: 1) 4. If A is a matrix of order 3X3 such that A (adjA) = 5I, then find IadjAI. (Ans: 25) 5. Find diag[1 23] + diag[456]. 6. Write number of matrices of order 2 x 3 with each entry 0, 1 or 2. 7. Write number of non-zero matrices of order 2 x 3 with each entry 0, 1 or 2. 8. Write number of non-zero and non-unit matrices of order 3x 3 with each entry 0, 1 or 2.

9. Write an example of matrix of order 3X3 a) Which is non-zero and symmetric b) which is non-zero and skew symmetric c) which is both symmetric and skew-symmetric 02 3 10. Find k and ������ if A =[ −2 2������ − 1 −5 ] is skew symmetric. −3 5 ������2 − ������ − 12 1 1 −2 11. Express [−1 4 3]as sum of symmetric and skew symmetric matrices. 2 −3 2 12. If f(x) = x2 +4x+5 and A =[14 −23], then find f (A). 13. For what value of x, if [31 + ������ 78] is (i) singular matrix (ii) non-singular matrix. − ������ 14. Find the determinant of diag [k 3k 5k] 15. If A = [02 00] , then find A2020 16. If A is a non-singular matrix such that A-1 = [ 15 531] , then find (AT)-1 −2 17. If A is a square matrix of order 3x3 such that IAI =2, then find IA.adj A I. ������ ������������������������ ������������������������ 18. If |−������������������������ −������ 1 | = 8, then find the value of x. ������������������������ 1 ������ 19. If matrix A = [−11 −11] and A2 = kA, then find the value of k. 20. (i) If M [21 09] = [49 63] , find matrix M (ii) If [21 09] M =[49 36], find matrix M 21. If [13 24] M[25 90] = [94 36] , find matrix M. −4 4 4 1 −1 1 22. Given that A = [−7 1 3 ] and B = [1 −2 −2] , find AB. Use this result to solve the 5 −3 −1 213 following system of linear equations: i) x – y + z = 4 ii) -4x -7y +5z = 4 x – 2y -2z = 9 4x + y – 3z = 9 2x + y +3z = 1 4x + 3y – z = 1 1 −1 2 −2 0 1 23. Use product [0 2 −3] [ 9 2 −3 ] to solve the system of equations 3 −2 4 6 1 −2

i) x – y + 2z = 1 ii) -2x + z =1 iii) -2x +9y +6z = 1 2y – 3z = 1 9x+2y-3z = 1 2y + z = 1 3x – 2y +4z =2 6x + y – 2z = 2 x – 3y – 2z = 2 31 2 24 . If A = [3 2 −3] , find A-1, Hence, solve the system of equations: 2 0 −1 3x + 3y +2z =1 x + 2y = 4 2x – 3y – z =5 −2 1 −7 −1 [−3 (Ans: A-1 = −7 15] and x = 2 , y = 1 , z = -4) 17 −4 2 3 25. An amount of Rs.5000 is put into three investments at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is Rs.358.If the combined income from the first two Investments is Rs.70 more than the income from the third, find the amount of each Investment by matrix method.(Ans: X = 1000 Y =2200 Z = 1800) 26. Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award Rs.x each, Rs.y each and Rs. z each For the three respectively values to its 3, 2 and 1 students with a total award money of Rs.1000.School Q wants to spend Rs.1500to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before).If the total amount of awards for one Prize on each value is Rs.600, using matrices, find the award money for each value. (Ans: x = 100, y = 200, z = 300) 27. Let A = [−14 −2 53] and B = 2 3 2 [4 2 5] .If BA = ( bij),then find the value of b21 +b32 . 1 28. If A = [−������������������������������������������������ ������������������������������������������������] , find α satisfying 0 < α <���2��� when A + AT = √2 I. 102 29. If A = [0 2 1] and A3 - 6A2 +7A+kI = 0 , then find K. 203 30. If A = [02 −35] and kA = [−08 45������������] , find the values of k , a and b.