Chapter 1.5 Inverses-LinearSystems

A if b is an n n matrix satisfying ba in then b a1

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Unformatted text preview: n × n matrix. (a) If B is an n × n matrix satisfying BA = In , then B = A−1 . (b) If B is an n × n matrix satisfying AB = In , then B = A−1 . Proof. Suppose B satisfies BA = In . Suppose c is a solution of the homogeneous system Ax = 0, so Ac = 0. Then c = In c = BAc = B 0 = 0 (the trivial solution). Then A is invertible, so B = BAA−1 = In A−1 = A−1 . To proof (b), consider B x = 0. Outline Number of Solutions Solving Multiple Systems More Results On the Inverse Properties of an Invertible Matrix Theorem Supp...
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