Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. Let us consider a map $f:M_n$ $\longrightarrow$ $M_n$ defined as $f_C(X)=X^2+C$, $C$ $\in$ $M_n$. Here $X^2=X*X$, where $'*'$ denotes the $q$-modulo matrix multiplications and $'+'$ is also standing for $q$-modulo addition.
I am interested to find out fixed points of the maps.
So I want to solve $f_C(X)=X$. It turns out a quadratic equation: $X^2+(-X)+C=0$.
Question is how many solutions are there of the equation in $M_n$? What are the solutions?
The algebraic matrix Riccati equation is given by $$ XBX+XA-DX-C=0 $$ with given matrices $A,B,C,D$ over a field $K$. For $B=I_n$, $A=-I_n$ and $D=0$ we obtain the equation $X^2-X-C=0$, which you want. All information about solutions and other things can be found in the book of P. Lancaster, L. Rodman: Algebraic Riccati equations. Clarendon Press, Oxford (1995). One method is based on so-called Jprdan-chains.