We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired M-matrix solution of the quadratic matrix equation X2 - EX - F -= 0 by connecting it with the nonsymmetric algebraic Riccati equation, where E is a diagonal matrix and F is an M-matrix.