Consider a polynomial $X^d+X^{d-1}+...+C=0$ where $X$ is a matrix whose entries are from the finite field $F_p$ and $C$ is also a matrix from $F_p$. How to verify the equation has roots in $F_p$. If there are roots, how to find them.
Any concrete references are also desired.
The existence of roots is a serious issue, as already the equation $X^2+X+C=0$ shows, for $d=2$ with $C=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ in $M_2(\mathbb{F}_p)$. It has no solution for $p=3$, but has one for $p=2$, for example. In general it has a solution over $\mathbb{F}_p$ if and only if the equation $(2t^2+t+1)(2t^2-t+1)=0$ has a solution in $\mathbb{F}_p$. This follows by a direct computation.
I don't know how to answer the question in general for arbitrary $d$ and $C$. Some useful information may be found in the following book:
-- F. R. Gantmacher: The theory of matrices. AMS Chelsea Publishing, Providence, RI (1998).