  We study some sum-product problems over matrix rings. Firstly, for $A, B,
C\subseteq M_n(\mathbb{F}_q)$, we have $$ |A+BC|\gtrsim q^{n^2}, $$ whenever
$|A||B||C|\gtrsim q^{3n^2-\frac{n+1}{2}}$. Secondly, if a set $A$ in
$M_n(\mathbb{F}_q)$ satisfies $|A|\geq C(n)q^{n^2-1}$ for some sufficiently
large $C(n)$, then we have $$ \max\{|A+A|, |AA|\}\gtrsim
\min\left\{\frac{|A|^2}{q^{n^2-\frac{n+1}{4}}}, q^{n^2/3}|A|^{2/3}\right\}. $$
These improve the results due to The and Vinh (2020), and generalize the
results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a
recent result due to The and Vinh (2020). Our method is based on spectral graph
theory and linear algebra.
