Initiated by Mulmuley, Vazirani, and Vazirani (1987), many algebraic algorithms have been developed for matching and related problems. In this paper, we review basic facts and discuss possible improvements with the aid of fast computation of the characteristic polynomial of a matrix. In particular, we show that the so-called exact matching problem can be solved with high probability in asymptotically the same time order as matrix multiplication. We also discuss its extension to the linear matroid parity problem.