# Are these matrices similar

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!

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