# C3091418 - 18 Let A and B be n n matrices and let C = AB...

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18. Let A and B be n × n matrices and let C = AB . Prove that if B is singular then C must be singular. Proof. Suppose that B is singular. By the part “(b) (a)” of Theorem 1.4.2, ( why this part? ) the equation B x = 0 has a nontrivial solution. That is, there is a vector x 0 6 = 0 such that B x 0 =
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Unformatted text preview: . Hence, AB x = A ( B x ) = A = . It means that the equation AB x = has a nontrivial solution x 6 = . By the part “(a) ⇒ (b)” of Theorem 1.4.2, ( again, why this part? ) C = AB is singular. / 1...
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