On a two-term recurrence for the determinant of a general matrix

Recently, a two-term recurrence for computing the determinant of a tridiagonal matrix has been found by El-Mikkawy [M. El-Mikkawy, A note on a three-term recurrence for a tridiagonal matrix, Appl. Math. Comput. 139 (2003) 503–511]. Then, the result has been extended to a block-tridiagonal matrix by Salkuyeh [D.K. Salkuyeh, Comments on “A note on a three-term recurrence for a tridiagonal matrix”, Appl. Math. Comput. 176 (2006) 442–444]. In this note, we show that the relation can be obtained for a general matrix and that as a by-product we obtain a generalization of the DETGTRI algorithm [M. El-Mikkawy, A fast algorithm for evaluating nth order tri-diagonal determinants, J. Comput. Appl. Math. 166 (2004) 581–584].