A new simple criterion is presented to verify if a matrix is positive definite, an \(M\)-matrix, or a Hurwitz matrix. It is based on the Gauss row elimination using inner and sign-preserving row operations. The computation of only one determinant is necessary. A sign-preserving row operation for the determinant means: (a) to multiply one row by a number \(k>0\); or (b) the same as (a) and add the result to any other row. An inner operation means: \(k\) rows of the resulting matrix are related only to the first \(k\) rows after the application of row operations on the original matrix. An illustrative numerical example is supplied.