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 $CD$.
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 $CD$.
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.