Eigenvalues of a matrix whose square is zero Let $A$ be a nonzero $3 \times 3$ matrix such that $A^2=0$. Then what is the number of non-zero eigenvalues of the matrix?
I am unable to figure out the eigenvalues of the above matrix.
P.S.:  how would the answer change if it were given that $A^3=0$?

A square matrix $A$ is called nilpotent if there is a $p \in \mathbb{N}$ such that $A^p=0$. So let $A$ be a nilpotent matrix. Then we have by definition of an eigenvalue
$$Av=\lambda v,$$
where $\lambda$ is an eigenvalue of $A$ and $v\neq 0$ is an eigenvector of $A$ to the corresponding eigenvalue. Because $A$ is nilpotent we also have
$$0=A^p v=\lambda^p v$$
and because $v \neq 0$ it follows $\lambda^p=0$, i. e. $\lambda=0$. So to your question: The number of non zero eigenvalues is in this case $0$.
