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# c1s5 - 1.5 Inverses of Matrices and Linear Systems 1...

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1.5 Inverses of Matrices, and Linear Systems 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.5. Inverses of Square Matrices Definition 1.15. An n × n matrix A is invertible if there exists an n × n matrix C such that AC = CA = I . If A is not invertible, it is singular . Theorem 1.9. Uniqueness of an Inverse Matrix. An invertible matrix has a unique inverse (which we denote A 1 ). Proof. Suppose C and D are both inverses of A . Then ( DA ) C = IC = C and D ( AC ) = D I = D . But ( DA ) C = D ( AC ) (associativity), so C = D . QED Example. It is easy to invert an elementary matrix. For example, sup- pose E 1 interchanges the first and third row and suppose E 2 multiplies row 2 by 7. Find the inverses of E 1 and E 2 .

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1.5 Inverses of Matrices, and Linear Systems 2 Theorem 1.10. Inverses of Products. Let A and B be invertible n × n matrices. Then AB is invertible and ( AB ) 1 = B 1 A 1 . Proof. By associativity and the assumption that A 1 and B 1 exist, we have: ( AB )( B 1 A 1 ) = [ A ( BB 1 )] A 1 = ( A I ) A 1 = AA 1 = I .
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c1s5 - 1.5 Inverses of Matrices and Linear Systems 1...

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