Characteristic polynomial/0 5 1 0/Eigenvalues/Example

We consider the real matrix ${\displaystyle {}M={\begin{pmatrix}0&5\\1&0\end{pmatrix}}}$. The characteristic polynomial is

${\displaystyle {}{\begin{aligned}\chi _{M}&=\det {\left(xE_{2}-M\right)}\\&=\det {\left(x{\begin{pmatrix}1&0\\0&1\end{pmatrix}}-{\begin{pmatrix}0&5\\1&0\end{pmatrix}}\right)}\\&=\det {\begin{pmatrix}x&-5\\-1&x\end{pmatrix}}\\&=x^{2}-5.\end{aligned}}}$

The eigenvalues are therefore ${\displaystyle {}x=\pm {\sqrt {5}}}$ (we have found these eigenvalues already in example, without using the characteristic polynomial).