Given an $n$ by $n$ matrix $A$ over a ring $F$ of prime characteristic $p$, the Charpoly( A, x ) mod p calling sequence computes the characteristic polynomial of $A$, a monic polynomial in $x$ of degree $n$ over $F$.

For matrices over the prime field $\mathrm{GF}\left(p\right)$, for $p$ a prime, Maple uses an $\mathrm{O}\left(n^3\right)$ algorithm. Otherwise, Maple uses an $\mathrm{O}\left(n^4\right)$ division free algorithm.

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

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

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

$x^3+x+2$

$p≔3\:$$\mathrm{alias}\left(a=\mathrm{RootOf}\left(x^2+1\right)\mathbf{mod}p\right)\:$

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

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

$C≔\mathrm{Charpoly}\left(A\,x\right)\mathbf{mod}p$

$C≔x^3+2\left(2+a\right)x^2+2\left(1+a\right)x+a$

$\mathrm{Factor}\left(C\right)\mathbf{mod}p$

$\left(x+2+2a\right)\left(x+2a\right)\left(x+a+2\right)$

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

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

$\mathrm{Charpoly}\left(A\,x\right)\mathbf{mod}2$

$x^3+x^2+\frac{t^6+1}{t^3}$