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assign8_soln

# assign8_soln - Math 136 Assignment 8 Solutions 1 For each...

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Math 136 Assignment 8 Solutions 1. For each of the following matrices, find the inverse, or show that the matrix is not invertible. a) A = 2 1 4 4 Solution: We have A - 1 = 1 2(4) - 1(4) 4 - 1 - 4 2 = 1 4 4 - 1 - 4 2 . b) B = 1 5 - 3 1 3 1 - 1 - 4 1 Solution: Row reducing [ B | I ] gives 1 5 - 3 1 0 0 1 3 1 0 1 0 - 1 - 4 1 0 0 1 1 5 - 3 1 0 0 0 - 2 4 - 1 1 0 0 0 0 3 / 2 1 / 2 1 Since the RREF of B is not I , B is not invertible. 2. Let A = 1 2 3 1 2 4 1 - 1 0 and B = 1 2 3 6 2 2 - 1 1 0 . a) Find det A and det B . Solution: We have det A = 1 2 3 1 2 4 1 - 1 0 = 1 3 3 1 3 4 1 0 0 = 3 det B = 1 2 3 6 2 2 - 1 1 0 = 3 2 3 8 2 2 0 1 0 = 18 b) Find det( AB ). Solution: det( AB ) = det A det B = 54 c) Find det( A + B ). Solution: det( A + B ) = 2 4 6 7 4 6 0 0 0 = 0 1

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2 d) Find A - 1 . Solution: We have 1 2 3 1 0 0 1 2 4 0 1 0 1 - 1 0 0 0 1 1 0 0 4 / 3 - 1 2 / 3 0 1 0 4 / 3 - 1 - 1 / 3 0 0 1 - 1 1 0 Thus A - 1 = 4 / 3 - 1 2 / 3 4 / 3 - 1 - 1 / 3 - 1 1 0 . 3. Write each of the following matrices and their inverses as a product of elementary matrices.
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assign8_soln - Math 136 Assignment 8 Solutions 1 For each...

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