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

For the matrix

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

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

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

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