## Matrices 3 x 3 over Z5

Determine eigenvalues and the corresponding eigenvectors for the following matrix over the field $$\mathbb Z_5$$. Decide whether this matrix is diagonalizable:
$$\begin{pmatrix} 1 & 0 & 0 \\ 2 & 3 & 3 \\ 1 & 1 & 0 \\ \end{pmatrix}$$
$$\lambda_1=2$$, $$\mathbf x_{1}=c\cdot(0, 1, 3)^T$$; $$\lambda_2=\lambda_3 = 1$$, $$\mathbf x_2=c\cdot(1, 0, 1)^T$$, $$\mathbf x_3=c\cdot(1, 4, 0)^T$$;
It is diagonalizable. The diagonal form is e.g. $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \\ \end{pmatrix}$$