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matrices assignment 4 solutions

# matrices assignment 4 solutions - Solutions 3 2 Given the...

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Solutions 1. Given the matrix A = - 0 2 1 3 find the values of the real number k for which det( A kI ) = 0 where I = 1 0 0 1 . (Total 4 marks) 2. det( A kI ) = 0 k k - - - 1 2 3 = 0 (M1) k 2 – 3 k + 2 = 0 (M1) ( k – 2)( k – 1) = 0 k = 1, 2 (A2) (C4) [4] 3. (a) Find the values of a and b given that the matrix A = - - - - 4 7 6 3 5 4 5 8 a is the inverse of the matrix B = . 3 1 2 1 2 1 3 1 - - - b (b) For the values of a and b found in part (a), solve the system of linear equations x + 2 y – 2 z = 5 3 x + by + z = 0 x + y – 3 z = a – 1. (Total 4 marks) 4. (a) AB = I ( AB ) 11 = 1 a – 12 + 6 = 1, giving a = 7 (A1) (C1) ( AB ) 22 = 1 –16 + 5 b + 7 = 1, giving b = 2 (A1) (C1) (b) The system is BX = 6 0 5 where X = z y x . Then, X = A - - - - = 6 0 5 4 3 5 7 5 8 6 4 7 6 0 5 . (M1) Thus, x = –1, y = 2, z = –1. (A1) (C2) [4] 1

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5. Find the values of the real number k for which the determinant of the matrix + - - 1 3 2 4 k k is equal to zero. (Total 3 marks) 6. 1 2 3 4 + - - k k = 0 ( k – 4)( k + 1) + 6 = 0 (M1) k 2 – 3 k + 2 = 0 (M1) ( k – 2)( k – 1) = 0 k = 2 or k = 1 (A1) (C3) [3] 7. (a) Given matrices A , B , C for which AB = C and det A 0, express B in terms of A and C . (2) (b) Let A = - - 2 2 3 3 1 2 3 2 1 , D = - - - - - 5 4 7 9 7 13 3 2 4 and C = 10 7 5 . (i) Find the matrix DA ; (ii) Find B if AB = C . (3) (c) Find the coordinates of the point of intersection of the planes x + 2 y + 3 z = 5, 2 x y + 2z = 7 and 3 x – 3 y + 2 z = 10. (2) (Total 7 marks) 8. (a) Since det A 0, A –1 exists. (M1) Hence AB = C B = A –1 C (C1) 2 (b) (i) DA = 1 0 0 0 1 0 0 0 1 (A1) (ii) B = A –1 C = DC (M1) 2
= - 2 1 1 (A1) 3 (c) The system of equations is x + 2 y + 3 z = 5 2 x y + 2 z = 7 3 x – 3 y + 2 z = 10 or A z y x = C . (M1) The required point = (1, –1, 2). (A1) 2 [7] 9. If A = 2 4 4 x and B = 4 8 2 y , find 2 values of x and y , given that AB = BA . (Total 3 marks) 10. AB = + + + = 8 4 24 16 32 2 4 8 2 2 4 4 y xy x y x (A1) BA = + + + = 40 16 8 8 2 4 2 2 4 4 4 8 2 x y y x x y (A1) AB = BA 8 x + 16 = 24 and 4 y + 8 = 40 This gives x = 1 and

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matrices assignment 4 solutions - Solutions 3 2 Given the...

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