The MinimalPolynomial(A, t) command returns a polynomial in t that is the minimal polynomial of Matrix A.

The minimal polynomial of A is the polynomial of lowest degree that divides every polynomial which has A as a zero.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$A≔⟨⟨3\,0\,1⟩\left|⟨-1\,2\,-1⟩\right|⟨0\,0\,2⟩⟩$

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

$\mathrm{mp}≔\mathrm{MinimalPolynomial}\left(A\,x\right)$

$\mathrm{mp}≔x^2-5x+6$

$\mathrm{divide}\left(\mathrm{CharacteristicPolynomial}\left(A\,x\right)\,\mathrm{mp}\right)$

$\mathrm{true}$

$P≔\mathrm{unapply}\left(\mathrm{mp}\,x\right)$

$P≔x\mapsto x^2-5\cdot x+6$

$P\left(A\right)$

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