# Row equivalence of partitioned matrices

Given two $$(n\times n)$$-matrices $$A$$ and $$B$$ for which the blockmatrix $$[A B]$$ is row-equivalent to $$[IX]$$, how do I find what $$X$$ is equal to?

This is part of the theory on blockmatrices and the Invertible Matrix Theorem (in my class) but i don't see how to apply the IMT here.

Saying that $$M$$ is row equivalent to $$N$$ means that there exists an invertible matrix $$S$$ such that $$SM=N$$.
Since $$S\begin{bmatrix} A & B \end{bmatrix} = \begin{bmatrix} SA & SB \end{bmatrix}$$, you can conclude from $$SA=I$$ that $$S=A^{-1}$$, so $$X=SB=A^{-1}B$$.