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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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    They could be equal unless you have further information. For example, if $n \geq 2,$ it is easy to find matrices $A,B,C,D,$ each of order $3,$ and all different, so that $AB = CD = I.$2012-03-09
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    Suppose AB or CD cannot be equal to I2012-03-09
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    That's not restrictive enough. Let $A = I$, $B = 6I$, $C = 2I$, $D = 3I$.2012-03-09
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    @user996522 : That wa just one easy example. You need to have much more information befoe you can be sure that $AB \neq CD$.2012-03-09
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    You can't prove it, because it's not true in general. It's possible that you might argue, with a **lot** of machinery to make sense of things, the you will "generally" have $AB\neq CD$ (that is, that under a suitable definition of "arbitrarily pick four distinct matrices" which allows a real probabilistic interpretation, the probability that they are equal is extremely small; though I'm not sure if that cna be made sensible), but you can't prove it always happens, because it doesn't. For instance, if $A$, $A^2$, $A^3$, and $A^4$ are pairwise distinct, you still get $AA^4 = A^2A^3$.2012-03-09
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    These matrices are not randomly picked though....2012-03-09

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