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Assuming their product exists, I can prove that the rank of the product of two equal rank matrices is less than or equal to the rank of either of the initial matrices.

However I'm struggling to find an example of two matrices whose product has a rank that is less than and NOT equal to the minimum of either.

Is there a theoretical way to simplify my search for an explicit case of this?

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up vote 4 down vote accepted

This isn't the most general example of what you're talking about, but Nilpotent matrices can be of help.

For example, take $$ A=B= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \\ AB= [0]_{2} $$

Then $rank(A)=rank(B)=1,$ but their product, the zero matrix, has zero rank.

More generally, if $M$ is a nilpotent matrix with $M^k =0$, then $A=M^{k-1},B=M$ will satisfy your requirement.

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$$ \pmatrix{1&0\\0&0}\pmatrix{0&0\\0&1}=\pmatrix{0&0\\0&0}\;. $$

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Here's a hint for the easiest possible case: Take $A = B$; can you find a $2 \times 2$ matrix such that $A^2 = 0$?

More generally, this is possible for any choices for the size of $A,B$. We know that $ABx = 0$ if $x$ is in the null space of $A$. So to make the the null space of $AB$ even larger, find $A$ so that the null space of $A$ shares vectors with the range of $B$.

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