Chapter 1.4 InverseMatrix

In particular the inverse of a is unique proof we

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Unformatted text preview: In particular, the inverse of A is unique. Proof. We have AB = In = BA and AC = In = CA. Therefore, C = CIn = C (AB ) = (CA)B = In B = B If A is an invertible n × n matrix, we will denote its unique inverse by A−1 . Thus we have: A−1 A = In = AA−1 . Outline Identity Matrices Inverse of a Matrix Inverses of 2 × 2 Matrices Theorem ab , then A is invertible if and only if ad − bc = 0, cd and in this case: If A = A−1 = Proof. Check. 1 ad − bc d −b −c a Outline Identity Matrices Inverse of a Matrix Inverses of Products Theorem If A and B are invertible n × n matrices, then AB is invertible with inverse (AB )−1 = B −1 A−1 Proof. (AB )−1 = B −1 A−1 means that AB has inverse B −1 A−1 . Check! Outline Identity Matrices Inverse of a Matrix Powers of Matrices Let r be a positive int...
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