C3092310 - A (adj A ) = (det A ) I to get A-1 (adj A-1 ) =...

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10. (Statement of the problem.) Proof. By the formula A (adj A ) = (det A ) I , we have (det A ) - 1 A (adj A ) = I. By Problem 15 in Section 2.2, adj A is nonsigular and (adj A ) - 1 = (det A ) - 1 A. Since det( A - 1 ) = (det A ) - 1 ( why? ), we get (adj A ) - 1 = (det A - 1 ) A. This is the first equality. We replace A by A - 1 in the formula
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Unformatted text preview: A (adj A ) = (det A ) I to get A-1 (adj A-1 ) = (det A-1 ) I. Multiplying by A from the left, we get AA-1 (adj A-1 ) = (det A-1 ) AI. That is, (adj A-1 ) = (det A-1 ) A. This is the second equality. / 1...
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