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 $SM=N$.
Indeed, each row operation is equivalent to multiplication on the left by an elementary matrix and, conversely, every invertible matrix is the product of elementary matrices.
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 $X=SB=A^{-1}B$.