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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 .
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
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