If every eigenvalue of $A$ is zero, does this mean $A$ is a zero matrix?

If every eigenvalue of $A$ is zero, show that $A$ is nilpotent. I got this question as my homework. I am just wondering if every eigenvalue of $A$ is zero, then $A$ is zero, why bother to prove $A$ is nilpotent.

Solution 1:

No, any strictly upper triangular matrix, such as:

$$\begin{pmatrix}0&1\\0&0\end{pmatrix}$$

will have all eigenvalues zero.

Solution 2:

MDP wrote that triangular matrix has all zero eigenvalues. The thing is that this is just a corollary from a more general statement.

Any nilpotent matrix has all zero eigenvalues

Nilpotent is the matrix which for some $k$ has $P^k = 0$.

For example for this non-obvious matrix is:

This can be proved by finding eigenvalue decomposition of $P^k$