It is given that
$$\begin{pmatrix}A & B \\ C & D \end{pmatrix} = \begin{pmatrix}A & 0 \\ C & I\end{pmatrix}\begin{pmatrix} I & A^{-1}B \\ 0 & D - CA^{-1}B\end{pmatrix}.$$
I would like to know if there is a rigorous precudure or algorithm in linear algebra to derive this form just from $\begin{pmatrix}A & B \\ C & D \end{pmatrix}$?
Let $A,B,C,D$ have size $n\times n$. Start from $$\begin{pmatrix}A&B\\C&D\end{pmatrix}.$$ Consider multiplication on the left by $$ \begin{pmatrix}A^{-1}&0\\0&I\end{pmatrix}. $$ This involves elementary row operations perfored to first $n$ rows to transform $A$ into $I$. After this, the matrix becomes $$\begin{pmatrix}I&A^{-1}B\\C&D\end{pmatrix}.$$ Then consider multiplication on the left by $$ \begin{pmatrix} I&0\\-C&I\end{pmatrix}. $$ This involves elementary row operations multiplying appropriate constants to first $n$ rows and subtracting from last $n$ rows. After this, the matrix becomes $$ \begin{pmatrix}I&A^{-1}B\\0&D-CA^{-1}B\end{pmatrix}. $$ Finally, $$\left[\begin{pmatrix}I&0\\-C&I\end{pmatrix} \begin{pmatrix}A^{-1}&0\\0&I\end{pmatrix} \right]^{-1} = \begin{pmatrix}A&0\\C&I\end{pmatrix}.$$