True/False: Matrix $\ \begin{pmatrix}1&3\\3&4\end{pmatrix}$ in $\mathbb{Z}_5$ is diagonalizable.
I have come to the conclusion that all the Eigenvectors for this matrix are not linearly independent, therefore it is not diagonalizable; however, I am not sure if my conclusion is correct. Would you be able to assist me in confirming my results? Thank you!
P.S. I also see that this matrix is symmetric, which should mean that it is diagonalizable, but maybe not in Zmod5?