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Suppose I consider the set of all matrices in $\mathrm{GL}(n, \mathbb{R})$, and I arbitrarily pick four distinct matrices $A,B,C,D$.

How can one prove that $AB$ is not equal to $CD$.

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    These matrices are not randomly picked though....2012-03-09

1 Answers 1

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Note that $AB=CD$ is equivalent to $D= C^{-1}AB$. If you pick four matrices at random, there are infinitely many choices for $D$ but only one of them will fail

$AB \neq CD \,.$

This shows that $AB \neq CD$ almost surely.