Minimal polynomial/2x2/Upper triangular matrix/Nilpotent/Example

For the matrix

${}M={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\,,$

${}X^{2}$ is the minimal polynomial. This polynomial becomes, after the substitution, the zero mapping, because of

${}M^{2}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}\,.$

The factors of ${}X^{2}$ of smaller degree are the constant polynomials ${}\neq 0$ and ${}a_{1}X$ with ${}a_{1}\neq 0$, but these polynomials do not annihilate ${}M$.