Define a new operation on matrices. It is called factorial. It is defined like a scalar on a matrix. (e.g.)
\[
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
!
=
\begin{pmatrix}
a! & b! \\
c! & d! \\
\end{pmatrix}
\]
note: the order of operations would be before multiplication and division.
This new operation works under addition.
\[
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
!
+
\begin{pmatrix}
e & f \\
g & h \\
\end{pmatrix}
!
=
\begin{pmatrix}
a! + e! & b! + f! \\
c! + g! & d! + h! \\
\end{pmatrix}
\]
This new operation works under multiplication.
\[
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
!
*
\begin{pmatrix}
e & f \\
g & h \\
\end{pmatrix}
!
=
\begin{pmatrix}
a!e! + b!g! & a!f! + b!h! \\
c!d! + d!g! & c!f! + d!h! \\
\end{pmatrix}
\]