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, ±nd the inverse, or show that the matrix is not invertible. a) A = 1 - 12 315 223 . Solution: To determine if A is invertible we write [ A | I ] and row reduce: 1 - 12100 3 1 5 0 1 0 2 2 3 0 0 1 1 - 1 2 1 0 0 04 - 1 - 310 0 0 0 1 - 11 Since the RREF of A will not be I , it follows that A is not invertible. b) B = 1 - 102 0 1 1 0 2 - 235 1 0 1 3 . Solution: To determine if E is invertible we write [ E | I ] and row reduce: 1 - 1021000 0 1 1 0 0 1 0 0 2 - 2350010 1 0 1 3 0 0 0 1 1000 10 3 8 3 - 1 3 - 5 3 0100 1 3 2 3 - 1 3 1 3 0010 - 1 3 1 3 1 3 - 1 3 0001 - 1 - 1 0 1 . Since the RREF of E is I , it follows that E is invertible and E - 1 = 10 3 8 3 - 1 3 - 5 3 1 3 2 3 - 1 3 1 3 - 1 3 1 3 1 3 - 1 3 - 1 - 1 0 1 2. Let B = 2 - 0 1 1 1 - 1 - 1 . Find B - 1 and use it to solve B±x = ± d , where ± d = (4 , - 2 , 3). Solution: To ±nd the inverse of B we write [ B | I ] and row reduce: 2 - 1 1 1 0 0 0 1 1 0 1 0 1 - 1 - 1001 1 0 0 0 1 1 010 - 1 / 23 / 21 0 0 1 1 / 2 - 1 / 2 - 1 Therefore B - 1 = 0 1 1 - 1 / / 1 / 2 - 1 / 2 - 1 . We have ±x = B - 1 ( B±x )= B - 1 4 - 2 3 = 0 1 1 - 1 / / 1 / 2 - 1 / 2 - 1 4 - 2 3 = 1 - 2 0 . 1
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2 3. a) Prove that if A and B are n × n matrices such that AB is invertible, then A and B are invertible.
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assign8_soln - Math 136 Assignment 8 Solutions 1. For each...

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