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 \(x\).