C3092215 - A is nonsingular if there is a matrix B such...

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10. (Statement of the problem.) Proof. Since AB = I , det( AB ) = det( I ). Since det( I ) = 1, by Theorem 2.2.3, det( A ) det( B ) = 1. It follows that det( A ) 6 = 0. Hence, A is nonsingular, and A - 1 exists. By AB = I , we get A - 1 ( AB ) A = A - 1 ( I ) A . That is, ( A - 1 A )( BA ) = A - 1 A . Now we get BA = I . In the definition of a nonsigular matrix, we say that a matrix
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Unformatted text preview: A is nonsingular if there is a matrix B such that AB = I and BA = I. Now since AB = I implies BA = I , we only need one of the equations in the definition, and the other follows automotically. / 1...
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