This paper describes a backtracking algorithm for the enumeration of nonisomorphic minimally nonideal (n2n) matrices that are not degenerate projective planes. The application of this algorithm for nh12 yielded 20 such matrices, adding 5 matrices to the 15 previously known. For greater dimensions, n=14 and n=17, 13 new matrices are given. For nonsquare matrices, 38 new minimally nonideal matrices are described.