## Eigenvalues of a matrix 5 x 5

Determine eigenvalues of the matrix

$$\begin{pmatrix} 3 & 2 & 0 & 1 & -2 \\ 0 & 2 & 0 & 0 & 0 \\ 2 & 0 & 1 & 0 & 3 \\ 0 & 5 & 0 & 1 & 0 \\ 4 & 8 & 0 & 7 & -3 \\ \end{pmatrix}$$.

• #### Resolution

By the expansion along the 2nd and the 4th row, and by the 3rd column we get

$$\begin{vmatrix} 3-t & 2 & 0 & 1 & -2 \\ 0 & 2-t & 0 & 0 & 0 \\ 2 & 0 & 1-t & 0 & 3 \\ 0 & 5 & 0 & 1-t & 0 \\ 4 & 8 & 0 & 7 & -3-t \\ \end{vmatrix}= (2-t)(1-t)^2 \begin{vmatrix} 3-t & -2 \\ 4 & -3-t \\ \end{vmatrix}=(2-t)(1-t)^3(1+t)$$

• #### Result

The eigenvalues are $$2$$, $$1$$ (multiplicity 3) a $$-1$$.