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# assign9 - Math 136 Assignment 9 Due Wednesday Mar 31st 1...

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Math 136 Assignment 9 Due: Wednesday, Mar 31st 1. Calculate the determinant of the following matrices. a) A = [~ ~ ~1 il C)C=Hb~cH 1 3 b) B = -1 1 1 d) D = [1 ~ i 1 - i -1 5 1 2 -3 8 1 2 2 -1 -1 -~ -1 + 2i [ a+ p 2. a) Prove that det ~ b+q e h c + r] [a b C] [P f = det d e f + det d k 9 h k 9 5 2 4 6 o -2 -1 1 1 0 q r] e f . h k [ a + P b + q C + r] b) Use part a) to express det d + x e + y f + z as the sum of determinants of 9 h k matrices who entries are not sums. 3. n x n matrices A and B are said to be similar if there exists an invertible matrix P such that p- 1 AP = B. Prove that if A and B are similar, then det A = det b. [ 0 1 31 -52] 4. Determine the inverse of A = -2 -6 7 by the cofactor method. 5. Use Cramer's Rule to solve the following systems. a) 2XI+X2=1 b) 5XI+X2-X3=4 3XI + 7X2 = -2 9XI + X2 - X3 = 1 Xl - X2 + 5X3 = 2 6. Let A = [~ ! JJ f = [~:] and b = Hl] Assuming that A is invertible use Cramer's Rule to find the value of X2 in the solution of the equation Ax = b. 7. In each case either prove the statement or give an example showing that it is false.

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