I suggest you switch from the bmatrix environment to an array environment. The third matrix in the following screenshot has vertical lines after columns 3 and 6, and horizontal lines after rows 3, 6, and 9.

\documentclass{article}
\usepackage{amsmath} % for 'align*' env.
\begin{document}
\begin{align}
A &=
\begin{bmatrix}
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
g & h & i & a & b & c & a & a & a & a \
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
g & h & i & a & b & c & a & a & a & a \
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
g & h & i & a & b & c & a & a & a & a
\end{bmatrix} \
&=
\left[ \begin{array}{@{} {10}{c} @{}} % switch to an 'array' setup
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
g & h & i & a & b & c & a & a & a & a \
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
g & h & i & a & b & c & a & a & a & a \
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
g & h & i & a & b & c & a & a & a & a
\end{array} \right] \
&\approx
\left[ \begin{array}{@{} ccc | ccc | cccc @{}} % add vertical and horizontal lines
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
g & h & i & a & b & c & a & a & a & a \
\hline
a & b & c & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
a & b & c & a & b & c & a & a & a & a \
\hline
d & e & f & a & b & c & a & a & a & a \
g & h & i & a & b & c & a & a & a & a \
a & b & c & a & b & c & a & a & a & a \
\hline
d & e & f & a & b & c & a & a & a & a \
d & e & f & a & b & c & a & a & a & a \
g & h & i & a & b & c & a & a & a & a
\end{array} \right]
\end{align*}
\end{document}