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Let $B \in M_n$ and let $A= \begin{bmatrix} B & C \\ 0 & 0 \\ \end{bmatrix} \in M_{n+m}$ such that $Rank([B \space \space C]) = Rank(B)$. Show that $A$ is similar to $B \oplus 0_m$

So since we have the rank condition it seems that $A$ can be reduced to $\begin{bmatrix} B & 0 \\ 0 & 0 \\ \end{bmatrix}$ via row and column operations. However I don't see how they necessarily need to be similar. Any help is appreciated thanks!

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Each column of $C$ must be a linear combination of the columns of $B$. That says $C = BS$ (where the $k$'th column of $S$ gives the coefficients for writing the $k$'th column of $C$ as a linear combination of the columns of $B$). Then

$$ \pmatrix{B & C\cr 0 & 0\cr} = \pmatrix{B & 0\cr 0 & 0\cr} \pmatrix{I & S\cr 0 & I\cr} = \pmatrix{I & -S\cr 0 & I\cr} \pmatrix{B & 0\cr 0 & 0\cr} \pmatrix{I & S\cr 0 & I\cr}$$ where $$\pmatrix{I & -S\cr 0 & I\cr} = \pmatrix{I & S\cr 0 & I\cr}^{-1}$$

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