Automorphism of a matrices ring 
Let $R$ be the ring of $3 \times 3$ matrices with coefficients in $\Bbb Z_5$.
  For every $g \in GL_3(\Bbb Z_5)$ prove that the function $$f\colon R \rightarrow R$$ defined as $$x \mapsto g^{-1}xg$$ is an automorphism of $R$. 

If I choose the matrix $$g = \begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{pmatrix},$$
how many matrices $x\in R$ such that $f(x)=x$ are there?
I have no idea how to solve this. Any help? 

This is much more general: suppose $R$ is a ring and $g$ is invertible in $R$. Then the map $f\colon R\to R$, $f(x)=gxg^{-1}$ is an automorphism of $R$.
Indeed, $f(x+y)=g(x+y)g^{-1}=gxg^{-1}+gyg^{-1}=f(x)+f(y)$ and
$$
f(xy)=gxyg^{-1}=gxg^{-1}gyg^{-1}=f(x)f(y)
$$
Obviously, $f(1)=1$.
Since the map $x\mapsto g^{-1}xg$ is the inverse of $f$, we are done.
For the second part, you want $gxg^{-1}=x$, so $gx=xg$. If
$$
x=\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
$$
then
$$
gx=\begin{pmatrix}
a_{31} & a_{32} & a_{33} \\
a_{21} & a_{22} & a_{23} \\
a_{11} & a_{12} & a_{13} \\
\end{pmatrix}
\qquad
xg=\begin{pmatrix}
a_{13} & a_{12} & a_{11} \\
a_{23} & a_{22} & a_{21} \\
a_{33} & a_{32} & a_{31}
\end{pmatrix}
$$
Thus you get
\begin{cases}
a_{31}=a_{13} \\
a_{32}=a_{12} \\
a_{33}=a_{11} \\
a_{21}=a_{23}
\end{cases}
and so the linear system has rank $4$.
