Recently Magnus and Neudecker [3] derived the dispersion matrix of vec X'X, when X' is a pxn random matrix (n>p) and vec X' has the distribution N np (vec M',I n GIV). This note is concerned with the matrix quadratic form X'AX, where X' is as defined above and A is a nonrandom (not necessarily symmetric) matrix. The dispersion matrix of vec X'AX will then be derived by applying results of Magnus and Neudecker [3] and Neudecker and Wansbeek [4]. It will be shown that an earlier partial and special result of Giguere and Styan [2] which assumes a symmetric A agrees with our result.