Find a matrix B such that $B^5 = A$ I am being asked to find a matrix $B$ where $B^5 = A$
$$A = \begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix}$$
In the first part of the question I was asked to find the eigenvalues & eigenvectors for the matrix which I found successfully.
If someone could help me finish this then that would be great.

HINT: If you have found the eigenvalues and eigenvectors, then you should easily be able to diagonalize this matrix as $A=PDP^{-1}$, where $D$ is diagonal. Then use the fact that $A^n=PD^n P^{-1}$, and the fact that the powers of a diagonal matrix are the matrices consisting of the powers of its entries.
