Non-zero nilpotent operator

Question

I know that if an operator $T\in L(V)$ is nilpotent, then $T^{dim(V)}=0$.

I am struggling to construct an example of a nilpotent operator $T\in L(V)$ which satisfies the condition $T^{dim(V)-1}\neq 0$.

Is there a such example?

It would be great if someone could explain in detail.

Answer 1

The companion matrix $C$ of $X^n$ has $X^n$ as its minimal polynomial. Therefore, $C^n=0$ but $C^{n-1}\ne0$.