The inverse and determinant of a given square matrix can be computed by the following routine applying simultaneously matrix order expansion and condensation. After completing the iteration, the expansion process results in the inverse of the given matrix (invM), and the condensation process generate an array of pivot elements (p) which eventualy gives the determinant (detM) of the given matrix (M):
[invM,detM,p,s,rc] = inv_det_opt(M).
In summary: (1) if the given matrix is non-singular then its determinant is found to be equal to the product of pivot elements. (2) if the last pivot element is found shrinking sharply toward zero, then the given matrix is said to be singular.