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quiz5-sol

# quiz5-sol - D = C-1 B-1 A-1 Then D ABC = C-1 B-1 A-1 A BC =...

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Name: MATH227 Quiz 5 (Fall 2006) For full credit, please show all steps in details!! 1. Suppose A is an n × n matrix and there exist n × n matrices C and D such that CA = I n and AD = I n . Prove that C = D . Is A invertible? Why? (5 points) Note that C = C ( I n ) = C ( AD ) = ( CA ) D = ( I n ) D = D. Thus, C = D . Now by the definition, since CA = I n and AC = AD = I n , A is invertible. 2. Suppose A , B , and C are invertible n × n matrices. Show that ABC is also invertible and find its inverse. (5 points) As A , B and C are invertible, A - 1 ,
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Unformatted text preview: D = C-1 B-1 A-1 . Then D ( ABC ) = C-1 B-1 ( A-1 A ) BC = C-1 B-1 ( I n ) BC = C-1 ( B-1 B ) C = C-1 ( I n ) C = C-1 C = I n and ( ABC ) D = AB ( CC-1 ) B-1 A-1 = AB ( I n ) B-1 A-1 = A ( BB-1 ) A-1 = A ( I n ) A-1 = AA-1 = I n . i.e., D ( ABC ) = I n = ( ABC ) D . Thus, ABC is invertible and D = C-1 B-1 A-1 is the inverse. 1...
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