Matrix equations often arise in chemical engineering mathematics (e.g., the Liapunov and Riccati equations) and it is important to have efficient methods for their solution. In this paper a new ＇＇elimination＇＇ method is proposed. The method uses the Cayley - Hamilton theorem to obtain relationships between a solution X to a matrix equation, the matrix coefficients in the equation and the characteristic coefficients of X. Given initial estimates of the characteristic coefficients it is then possible to formulate an iterative scheme to determine X itself. The method can be used more directly when the matrix equation reduces to finding the m(th) roots of a matrix Q, say, in which case interesting algebraic expressions linking the characteristic coefficients of Q and those of its roots can be obtained. Some illustrative examples are given and an attempt is made to compare the new approach with currently available alternatives.