The point of Gaussian elimination is to solve/identiy-ify a linear equation. Take, if you have a matrix expression:
\begin{equation} Ax = b \end{equation}
We can apply A^{-1} to both side, we then have:
\begin{equation} A^{-1}Ax = A^{-1} b \end{equation}
Applying the definition of the identity:
\begin{equation} Ix = A^{-1}b \end{equation}
Therefore, to solve for some A^{-1}, which would yield x.