With exception of the zero matrix, can a matrix be nilpotent $(A^k=0)$ and idempotent $(A^2=A)$ at the same time?
and
With exception of the identity matrix, can a matrix be idempotent and involutory $(A^2=I)$ at the same time?
I think the two answers are negative. Can anyone help me find a counterexample, if any?
Best Answer
Assume $A^2=A$ and there exists a minimal $k\in\Bbb N$ with $A^k=0$. If $k\ge 2$, then $0=A^k=A^2A^{k-2}=AA^{k-2}A^{k-1}$, contradicting minimality of $k$. Therefore $k=1$ and so $A=A^1=0$.
If $A^2=A$ and $A^2=I$, then $A=A^2=I$.