The Det function is a placeholder for representing the determinant of the matrix A.  It is used in conjunction with mod and modp1 which define the coefficient domain as described below.

The call $\mathrm{Det}\left(A\right)\mathbf{mod}m$ computes the determinant of the matrix $A\mathbf{mod}m$ in characteristic m which may not not be prime.  The entries in A may be integers, rationals, polynomials, or in general, rational functions in parameters over a finite field.

The call $\mathrm{modp1}\left(\mathrm{Det}\left(A\right)\,p\right)$ computes the determinant of the matrix $A\mathbf{mod}p$ where p is a prime integer and the entries of A are modp1 polynomials using fraction-free Gaussian elimination.

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

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

$\mathrm{Det}\left(A\right)\mathbf{mod}3$

$2$

$\mathrm{Det}\left(A\right)\mathbf{mod}6$

$5$

$C≔\mathrm{Matrix}\left(\left[\left[x-2\,3\,1\right]\,\left[3\,x-2\,3\right]\,\left[0\,3\,x-2\right]\right]\right)$

$C≔\left[\begin{array}{ccc}x-2& 3& 1\\ 3& x-2& 3\\ 0& 3& x-2\end{array}\right]$

$\mathrm{Det}\left(C\right)\mathbf{mod}3$

$x^3+1$

$\mathrm{Charpoly}\left(A\,x\right)\mathbf{mod}3$

$x^3+1$

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(x^4+x+1\right)\right)\:$

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

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

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

$\mathrm{\alpha }$

$A≔\mathrm{Matrix}\left(\left[\left[1-\mathrm{\alpha }\,\frac{\mathrm{\alpha }}{t}\,1-\mathrm{\alpha }t\right]\,\left[1+\mathrm{\alpha }\,\mathrm{\alpha }t\,1+\mathrm{\alpha }t\right]\,\left[\mathrm{\alpha }\,1-\frac{\mathrm{\alpha }}{t}\,\mathrm{\alpha }t\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1-\mathrm{\alpha }& \frac{\mathrm{\alpha }}{t}& -\mathrm{\alpha }t+1\\ 1+\mathrm{\alpha }& \mathrm{\alpha }t& \mathrm{\alpha }t+1\\ \mathrm{\alpha }& 1-\frac{\mathrm{\alpha }}{t}& \mathrm{\alpha }t\end{array}\right]$

$\mathrm{collect}\left(\mathrm{Det}\left(A\right)\mathbf{mod}2\,t\right)$

${\mathrm{\alpha }}^2{t}^2+{\mathrm{\alpha }}^{2}t+{\mathrm{\alpha }}^2+\frac{{\mathrm{\alpha }}^2}{t}$