We are concerned about various strongly clean properties of a kind of matrix subrings $L_{(s)}(R)$ over a local ring $R$. Let $R$ be a local ring, and let $s\in C(R)$. We prove that $A\in L_{(s)}(R)$ is strongly clean if and only if $A$ or $I_2-A$ is invertible, or $A$ is similar to a diagonal matrix in $L_{(s)}(R)$. Furthermore, we prove that $A\in L_{(s)}(R)$ is quasipolar if and only if $A\in GL_2(R)$ or $A\in L_{(s)}(R)^{qnil}$, or $A$ is similar to a diagonal matrix $\left( \begin{array}{cc} \lambda&0\\ 0&\mu \end{array} \right)$ in $L_{(s)}(R)$, where $\lambda\in J(R)$, $\mu\in U(R)$ or $\lambda\in U(R)$, $\mu\in J(R)$, and $l_{\mu}-r_{\lambda}$, $l_{\lambda}-r_{\mu}$ are injective. Pseudopolarity of such matrix subrings is also obtained.