A ring R is called quasipolar if for every a 2 R there exists $p^2=p{in}R$ such that $p{in}comm^2{_R}(a)$, $ a+p{in}U(R)$ and $ap{in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{ imes}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{ imes}2$ matrix ring to be quasipolar are obtained.