Characteristic polynomial/0 5 1 0/Eigenvalues/Example

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

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

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