Suppose $A,B$ are square matrices of the form $A = \left[C_{1}\middle|\frac{I}{0\dots0}\right], B= \left[C_{2}\middle|\frac{I}{0\dots0}\right]$.

Assume that $n$ is a multiple of $3$. For, e.g., $n=6$
$$\tag{1}\label{1}
A = \begin{pmatrix}
a_1 & 1 & 0 & 0 & 0 & 0 \\
a_2 & 0 & 1 & 0 & 0 & 0 \\
a_3 & 0 & 0 & 1 & 0 & 0 \\
a_4 & 0 & 0 & 0 & 1 & 0 \\
a_5 & 0 & 0 & 0 & 0 & 1 \\
a_n & 0 & 0 & 0 & 0 & 0
\end{pmatrix},\quad
A^{-1} = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 1/a_n \\
1 & 0 & 0 & 0 & 0 & -a_1/a_n \\