Answer
\[0,2,-1\]
Work Step by Step
\[A=\left[\begin{array}{ccc}0&0&0\\
0&2&5\\
0&0&-1\\\end{array}\right]\]
Characteristic polynomial of $A$ is given by
\[|A-\lambda I|=0\]
\[A-\lambda I=\left[\begin{array}{ccc}0&0&0\\
0&2&5\\
0&0&-1\\\end{array}\right]-\lambda\left[\begin{array}{ccc}1&0&0\\
0&1&0\\
0&0&1\\\end{array}\right]\]
\[A-\lambda I=\left[\begin{array}{ccc}-\lambda &0&0\\
0&2-\lambda &5\\
0&0&-1-\lambda \\\end{array}\right]\]
\[|A-\lambda I|=\left|\begin{array}{ccc}-\lambda &0&0\\
0&2-\lambda &5\\
0&0&-1-\lambda \\\end{array}\right|\]
\[|A-\lambda I|=(-\lambda)(2-\lambda)(-1-\lambda)\]
For eigen values of $A$
\[|A-\lambda I|=0\]
\[\Rightarrow \lambda_{1}=0\:,\:\lambda_2=2\:,\:\lambda_3=-1\]
Hence eigen values of $A$ are $0,2$ and $-1$.
