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Unformatted text preview: 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 = 1 1 2 3 1 5 2 2 3 . Solution: To determine if A is invertible we write [ A  I ] and row reduce: 1 1 2 1 0 0 3 1 5 0 1 0 2 2 3 0 0 1 ∼ 1 1 2 1 4 1 3 1 1 1 1 Since the RREF of A will not be I , it follows that A is not invertible. b) B = 1 1 0 2 1 1 0 2 2 3 5 1 1 3 . Solution: To determine if E is invertible we write [ E  I ] and row reduce: 1 1 0 2 1 0 0 0 1 1 0 0 1 0 0 2 2 3 5 0 0 1 0 1 1 3 0 0 0 1 ∼ 1 0 0 0 10 3 8 3 1 3 5 3 0 1 0 0 1 3 2 3 1 3 1 3 0 0 1 0 1 3 1 3 1 3 1 3 0 0 0 1 1 1 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 1 2. Let B = 2 1 1 1 1 1 1 1 . Find B 1 and use it to solve B~x = ~ d , where ~ d = (4 , 2 , 3). Solution: To find the inverse of B we write [ B  I ] and row reduce: 2 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 ∼ 1 0 0 1 1 0 1 0 1 / 2 3 / 2 1 0 0 1 1 / 2 1 / 2 1 Therefore B 1 = 1 1 1 / 2 3 / 2 1 1 / 2 1 / 2 1 . We have ~x = B 1 ( B~x ) = B 1 4 2 3 = 1 1 1 / 2 3 / 2 1 1 / 2 1 / 2 1 4 2 3 = 1 2 ....
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This note was uploaded on 05/04/2010 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.
 Spring '08
 All
 Linear Algebra, Algebra, Matrices

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