Certain strongly clean matrices over local rings

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 ∈ C(R) . We prove that A ∈ L(s)(R) is strongly clean if and only if A or I2−A is invertible, or A is similar to a diagonal matrix in L(s)(R) . Furthermore, we prove that A ∈ L(s)(R) is quasipolar if and only if A ∈ GL2(R) or A ∈ L(s)(R) , or A is similar to a diagonal matrix ( λ 0 0 μ ) in L(s)(R) , where λ ∈ J(R) , μ ∈ U(R) or λ ∈ U(R) , μ ∈ J(R) , and lμ − rλ , lλ − rμ are injective. Pseudopolarity of such matrix subrings is also obtained.