A $3 \times 3$ matrix with minimal polynomial $x^2$

Find a $3 \times 3$ matrix for which the minimal polynomial is $x^2$.

Solution

Take \[ A=\begin{bmatrix} 0&1&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}. \] Then $A\ne0$ but $A^2=0$. Hence $x^2$ annihilates $A$, while neither the constant polynomial $1$ nor the degree-one polynomial $x$ annihilates $A$. Therefore the minimal polynomial is \[ m_A(x)=x^2. \]