The Inverse function is a placeholder for representing the inverse of a square matrix A.

The call Inverse(A) mod n computes the inverse of the square matrix A over a finite ring of characteristic n. This includes finite fields, GF(p), the integers mod p, and GF(p^k) where elements of GF(p^k) are expressed as polynomials in RootOf's.

$A≔\mathrm{Matrix}\left(\left[\left[1\,2\,3\right]\,\left[1\,3\,0\right]\,\left[1\,4\,3\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& 2& 3\\ 1& 3& 0\\ 1& 4& 3\end{array}\right]$

$B≔\mathrm{Inverse}\left(A\right)\mathbf{mod}5$

$B≔\left[\begin{array}{ccc}4& 1& 1\\ 2& 0& 3\\ 1& 3& 1\end{array}\right]$

$\mathrm{`~`}\left[\mathrm{Expand}\right]\left(A·B\right)\mathbf{mod}5$

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

$\mathrm{Inverse}\left(A\right)\mathbf{mod}2$

Error, (in `mod/Inverse`) singular matrix

$\mathrm{alias}\left(a=\mathrm{RootOf}\left(x^4+x+1\right)\mathbf{mod}2\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,a\,a^2\right]\,\left[1\,a^2\,1\right]\,\left[1\,a^3\,a^2\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& a& a^2\\ 1& a^2& 1\\ 1& a^3& a^2\end{array}\right]$

$B≔\mathrm{Inverse}\left(A\right)\mathbf{mod}2$

$B≔\left[\begin{array}{ccc}{a}^2+a& a^3+a& a^3+a^2+1\\ a^3+a^2& 0& a^3+a^2\\ a+1& a^3+a+1& a^3\end{array}\right]$

$\mathrm{`~`}\left[\mathrm{Expand}\right]\left(A·B\right)\mathbf{mod}2$

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