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Every time I think I found a solution, it turns out that the rank of my matrices reduces so that rank(A) no longer equals rank(B).

I'm just guessing and checking, perhaps there is a more formulaic approach?

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$A=\left(\begin{array}{rr}% 1&0\\% 0&0\\% \end{array}\right)%$, $B:=\left(\begin{array}{rr}% 0&0\\% 1&0\\% \end{array}\right)%$

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  • $\begingroup$ Keep in memory these two matrices, they are common counter-examples in this kind of problem. $\endgroup$ – Traklon Nov 4 '14 at 8:41
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$$\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{pmatrix}^2=\begin{pmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}$$ $$\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}^2=\begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}$$

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