The command IsZeroMatrix(A, rc, R) returns true if and only if A is null modulo the saturated ideal of rc.

It is assumed that rc is strongly normalized.

This command is part of the RegularChains[MatrixTools] package, so it can be used in the form IsZeroMatrix(..) 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][IsZeroMatrix](..).

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

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

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc}≔\mathrm{Empty}\left(R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

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

$\mathrm{Equations}\left(\mathrm{rc}\,R\right)$

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

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

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

$\mathrm{IsZeroMatrix}\left(m\,\mathrm{rc}\,R\right)$

$\mathrm{false}$