1. 1. MODULE 12 MATEMATIK SPM ENRICHMENT TOPIC : MATRICES TIME : 2 HOURS 3 2 4 n 1 . (a) The inverse matrix of is m 5 4 5 3 Find the value of m and of n. (b) Hence, using matrices, solve the following simultaneous equations : 3x 2y = 8 5x 4y = 13 Answer : (a) (b) 2 . (a) Given that G = m 3 and the inverse matrix of G is 1 4 3 , 2 n 1 4 2 m find the value of m and of n. (2) Hence, using matrices, calculate the value of p and of q that satisfies the following equation : p 1
2. 2. G = q 8 Answer : (a) (b) 3 . (a) Given that 1 2 1 0 find matrix A. A = , 3 5 0 1 (2) Hence, using the matrix method, find the value of r and s which satisfy the simultaneous equations below. -r + 2s = -4 -3r + 5s = -9 Answer : (a)
3. 3. (b) 4 5 1 0 4 . Given matrix P = and matrix PQ = 6 8 0 1 (1) Find the matrix Q. (2) Hence, calculate by using the matrix method, the values of m and n that satisfy the following simultaneous linear equations : 4m + 5n = 7 6m + 8n = 10 Answer : (a) (b)
4. 4. 4 3 5. Given the matrix P is , 8 5 1 0 (a) Find the matrix Q so that PQ = 0 1 (b) Hence, calculate the values of h and k, which satisfy the matrix equation: 4 3h 7 = 8 5 k 1 1 Answer : (a) (b) k 6 6 . (a) Given matrix M = , find the value of k if matrix M has no inverse. 2
5. 5. 4 (b) Given the matrix equations 7 6 x 4 a n d x 1 8 6 4 = = 5 8 y 1 y h 5 7 1 (1) Find the value of h (2) Hence, find the value of x and y. Answer : (a) (b) 2 5 7 . It is given that matrix P = does not have an inverse matrix. k 2 (a) Find the value of k. (b) If k = 1, find the inverse matrix of P and hence, using matrices, find the values of x and y that satisfy the following simultaneous linear equations. 2x + 5y =
6. 6. 13 x - 2y = -7 Answer : (a) (b) 8 . (a) Find matrix M such that 2 4 2 4 M = 1 3 1 3 (2) Using matrices, calculate the values of x and y that satisfy the following matrix equation. 2 4 x 6 = 1 3 y 5 Answer : (a)
7. 7. (b)
8. 8. 9 . (a) Find the inverse of matrix 3 1 . 5 2 (2) Hence, using matrices, calculate the values of d and e that satisfy the following simultaneous equations : 2d e = 7 5d e = 16 Answer : (a) (b)
9. 9. 1 2 1 0. Given matrix M = , find 2 5 (a) the inverse matrix of M (b) hence, using matrices, the values of u and v that satisfy the following simultaneous equations : u 2v = 8 2u + 5v = 7 Answer : (a) (b) MODULE 12 - ANSWERS TOPIC : MATRICES 1 . ( a ) m = 1 1m 2 n = 2 1m ( b ) 3 2 x 8
10. 10. = 1 m 5 4 y 1 3 x 1 4 28 1m = y 5 2 3 1 3 x = 3 1m y = 1 1m 2 2 . ( a ) n = 4 1 m m = 5 1 m ( b ) 5 3 p 1 = 2 4 q 8 p 1 4 3 1 = q 1 4 2 5 8 p = 2 q = -3 5 2 3 . ( a ) A = 3 1 1 2 r 4 = (b ) 3 5 s 9 1 m 1 m 1 m 1 m 2 m 1 m r 1 5 2 = s 1 3 r = -2 1 m
11. 11. s = -3 81 4 . (a) P =
12. 12. 1 8 5 = 2 6 4 1 m 4 5m 7 (b ) = 1 m 6 8 n 1 0 m 1 8 5 7 = 1 m n 2 6 4 1 0 m = 3 1 m n = -1 1 m 5 . (a ) P 1 5 3 1 m = 20 (24) 8 4 1 5 3 1 m= 4 8 4 (b ) 4 3 h 7 = 8 5 k 1 1 h 1 5 3 7
13. 13. = 1 m k 2 8 4 1 1 1 2 = 1 m 2 1 00 h = 1 1 m k = -50 1 m 6 . (a ) k = -12 1 m (b ) ( i ) h = 26 1 m x 1 8 6 4 = y 2 6 5 7 1 1 26 = 2 6 1 3 ( i i ) x = -1 y = 1 2
14. 14. 7 . (a ) - 4 5k = 0 1 m 5k = -4 k = 4 1 m 5 (b ) 2 5 x 13 = 1 2 y 7 x 1 2 5 13 = y 9 1 2 7 x = -1 y = 3 8 . (a ) M = 1 0 0 1 (b ) x 1 3 46 = y 6 4 1 2 5 1 3 4 6 = 2 1 2 5 1 2 = 4
15. 15. 2 x = -1 y = 2 1m 1m 1m 1m 1m 1m 1m 1m 2m 1m 1m 1m 1m 9. ( a ) 1 2 1
16. 16. 6 + 5 5 3 1 2 1 = 1 5 3 ( b ) 2 1d 7 = 5 3 e 1 6 d 1 3 1 7 = e 1 5 2 1 6 1 5 = 31 5 = 3 d = 5 e = 3 10 . ( a ) 1 5 2 5 (4) 2 1
17. 17. 1 5 2 = 9 2 1 ( b ) 1 2u 8 = 2 5 v 7 u 1 5 28 = v 9 2 1 7 1 54 = 9 9 6 = 1 u = 6 v = 1 1m 1m 1m
18. 18. 1m 1m 1m 1m 1m 1m 1m 1m 1m