The MatrixPower function efficiently computes the nth power of the input mod m Matrix via binary powering.

This command is part of the LinearAlgebra[Modular] package, so it can be used in the form MatrixPower(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][MatrixPower](..).

$\mathrm{with}\left(\mathrm{LinearAlgebra}\left[\mathrm{Modular}\right]\right)\:$

$A≔\mathrm{Mod}\left(13\,\mathrm{Matrix}\left(\left[\left[2\,0\right]\,\left[0\,2\right]\right]\right)\,\mathrm{integer}\left[\right]\right)$

$A≔\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

$\mathrm{MatrixPower}\left(13\,A\,0\right)$

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\mathrm{MatrixPower}\left(13\,A\,1\right)$

$\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

$\mathrm{MatrixPower}\left(13\,A\,20\right)$

$\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right]$