The LeftDivision(A, B, x) command computes a left quotient Q and a remainder R such that $A=B·Q+R$ where $B^{\left(\mathrm{-1}\right)}·R$ is strictly proper.  That is,  $\underset{z\to \mathrm{\infty }}{lim}{B}^{\left(-1\right)}\left(z\right).R\left(z\right)$ is a zero matrix. The input matrices must have the same number of rows, and B must be a square nonsingular matrix of polynomials.

The RightDivision(A, B, x) command computes a right quotient Q and a remainder R such that $A=Q·B+R$ where $R·B^{\left(\mathrm{-1}\right)}$ is strictly proper.  That is,  $\underset{z\to \mathrm{\infty }}{lim}R\left(z\right).B^{\left(-1\right)}\left(z\right)$ is a zero matrix. The input matrices must have the same number of columns, and B must be a square nonsingular matrix of polynomials.

The quotient $Q$ and the remainder $R$ are returned in a list.

$\mathrm{with}\left(\mathrm{MatrixPolynomialAlgebra}\right)\:$

$A≔\mathrm{Matrix}\left(2\,2\,\left[\left[-9z^2-3z+1\,12z^2+10z\right]\,\left[-3z^3+2z^2-z\,4z^3+2z-2z^2\right]\right]\right)\:$

$B≔\mathrm{Matrix}\left(2\,2\,\left[\left[-3z^3+6z^2+5z+1\,-12z^2-13z\right]\,\left[z^4+z^3+z^2\,-4z^3-3z+3z^2\right]\right]\right)\:$

$Q,R≔\mathrm{op}\left(\mathrm{LeftDivision}\left(A\,B\,z\right)\right)$

$Q,R≔\left[\begin{array}{cc}0& 0\\ \frac{3}{4}& \mathrm{-1}\end{array}\right],\left[\begin{array}{cc}\frac{27z}{4}+1& -3z\\ -\frac{1}{4}{z}^2+\frac{5}{4}z& z^2-z\end{array}\right]$

$\mathrm{map}\left(\mathrm{expand}\,A-\left(B·Q+R\right)\right)$

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

$\mathrm{map}\left(f\mapsto \mathrm{limit}\left(f\,z=\mathrm{\infty }\right)\,\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(B\right)·R\right)$

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

$Q,R≔\mathrm{op}\left(\mathrm{RightDivision}\left(A\,B\,z\right)\right)$

$Q,R≔\left[\begin{array}{cc}-\frac{1}{2}& 0\\ -\frac{z}{6}+\frac{13}{36}& -\frac{1}{2}\end{array}\right],\left[\begin{array}{cc}-6z^2-\frac{1}{2}z+\frac{3}{2}-\frac{3}{2}{z}^3& 6z^2+\frac{7}{2}z\\ -\frac{5}{12}{z}^3+\frac{7}{6}{z}^2-\frac{95}{36}z-\frac{13}{36}& \frac{5}{3}{z}^2+\frac{187}{36}z\end{array}\right]$

$\mathrm{map}\left(\mathrm{expand}\,A-\left(Q·B+R\right)\right)$

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

$\mathrm{map}\left(f\mapsto \mathrm{limit}\left(f\,z=\mathrm{\infty }\right)\,R·\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(B\right)\right)$

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