The command MatrixInverse(A, rc, R) returns two lists.

The first list the command returns is a list of pairs $\left[B_i\,{\mathrm{rc}}_i\right]$ where ${\mathrm{rc}}_i$ is a regular chain and $B_i$ is the inverse of A modulo the saturated ideal of ${\mathrm{rc}}_i$.

The second list the command returns is a list of triplets $\left[\mathrm{noInv}\,A\,{\mathrm{rc}}_i\right]$ where ${\mathrm{rc}}_i$ is a regular chain and A is the input matrix such that A is not invertible modulo the saturated ideal of ${\mathrm{rc}}_i$.

All the returned regular chains ${\mathrm{rc}}_i$ form a triangular decomposition of rc (in the sense of Kalkbrener).

It is assumed that rc is strongly normalized.

The algorithm is an adaptation of the algorithm of Bareiss.

This command is part of the RegularChains[MatrixTools] package, so it can be used in the form MatrixInverse(..) only after executing the command with(RegularChains[MatrixTools]).  However, it can always be accessed through the long form of the command by using RegularChains[MatrixTools][MatrixInverse](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$$\mathrm{with}\left(\mathrm{MatrixTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[y\,z\right]\right)\;$$\mathrm{rc}≔\mathrm{Empty}\left(R\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[z^4+1\,y^2-z^2\right]\,\mathrm{rc}\,R\right)\:$$\mathrm{Equations}\left(\mathrm{rc}\,R\right)$

$\left[y^2-z^2\,z^4+1\right]$

$m≔\mathrm{Matrix}\left(\left[\left[1\,y+z\right]\,\left[0\,y-z\right]\right]\right)$

$m≔\left[\begin{array}{cc}1& y+z\\ 0& y-z\end{array}\right]$

$\mathrm{mim}≔\mathrm{MatrixInverse}\left(m\,\mathrm{rc}\,R\right)$

$\mathrm{mim}≔\left[\left[\left[\left[\begin{array}{cc}1& 0\\ 0& \frac{z^3}{2}\end{array}\right]\,\mathrm{regular\_chain}\right]\right]\,\left[\left[''noInv''\,\left[\begin{array}{cc}1& y+z\\ 0& y-z\end{array}\right]\,\mathrm{regular\_chain}\right]\right]\right]$

$\mathrm{m1}≔\mathrm{mim}\left[1\right]\left[1\right]\left[1\right]\;$$\mathrm{rc1}≔\mathrm{mim}\left[1\right]\left[1\right]\left[2\right]\;$$\mathrm{Equations}\left(\mathrm{rc1}\,R\right)$

$\mathrm{m1}≔\left[\begin{array}{cc}1& 0\\ 0& \frac{z^3}{2}\end{array}\right]$

$\mathrm{rc1}≔\mathrm{regular\_chain}$

$\left[y+z\,z^4+1\right]$

$\mathrm{MatrixMultiply}\left(\mathrm{m1}\,m\,\mathrm{rc1}\,R\right)$

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

$m≔\mathrm{Matrix}\left(\left[\left[1\,y+z\right]\,\left[2\,y-z\right]\right]\right)$

$m≔\left[\begin{array}{cc}1& y+z\\ 2& y-z\end{array}\right]$

$\mathrm{mim}≔\mathrm{MatrixInverse}\left(m\,\mathrm{rc}\,R\right)$

$\mathrm{mim}≔\left[\left[\left[\left[\begin{array}{cc}1& 0\\ -z^3& \frac{z^3}{2}\end{array}\right]\,\mathrm{regular\_chain}\right]\,\left[\left[\begin{array}{cc}0& \frac{1}{2}\\ -\frac{z^3}{2}& \frac{z^3}{4}\end{array}\right]\,\mathrm{regular\_chain}\right]\right]\,\left[\right]\right]$

$\mathrm{m1}≔\mathrm{mim}\left[1\right]\left[1\right]\left[1\right]\;$$\mathrm{rc1}≔\mathrm{mim}\left[1\right]\left[1\right]\left[2\right]$

$\mathrm{m1}≔\left[\begin{array}{cc}1& 0\\ -z^3& \frac{z^3}{2}\end{array}\right]$

$\mathrm{rc1}≔\mathrm{regular\_chain}$

$\mathrm{m2}≔\mathrm{mim}\left[1\right]\left[2\right]\left[1\right]\;$$\mathrm{rc2}≔\mathrm{mim}\left[1\right]\left[2\right]\left[2\right]$

$\mathrm{m2}≔\left[\begin{array}{cc}0& \frac{1}{2}\\ -\frac{z^3}{2}& \frac{z^3}{4}\end{array}\right]$

$\mathrm{rc2}≔\mathrm{regular\_chain}$

$\mathrm{MatrixMultiply}\left(\mathrm{m2}\,m\,\mathrm{rc2}\,R\right)$

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

$\mathrm{MatrixMultiply}\left(\mathrm{m2}\,m\,\mathrm{rc2}\,R\right)$

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

$\mathrm{clr}≔\mathrm{MatrixCombine}\left(\left[\mathrm{rc1}\,\mathrm{rc2}\right]\,R\,\left[\mathrm{m1}\,\mathrm{m2}\right]\right)$

$\mathrm{clr}≔\left[\left[\left[\begin{array}{cc}\frac{y{z}^3}{2}+\frac{1}{2}& -\frac{y{z}^3}{4}+\frac{1}{4}\\ \frac{1}{4}y{z}^2-\frac{3}{4}{z}^3& -\frac{1}{8}y{z}^2+\frac{3}{8}{z}^3\end{array}\right]\,\mathrm{regular\_chain}\right]\right]$

$\mathrm{MatrixMultiply}\left(\mathrm{clr}\left[1\right]\left[1\right]\,m\,\mathrm{clr}\left[1\right]\left[2\right]\,R\right)$

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