Qualitatively invertible matrices (Q798729)

The purpose of this work is to study the class of qualitatively invertible matrices: the class of matrices A such that, based only upon a qualitative analysis, it can be determined that \(\det\quad A\neq 0.\) For irreducible matrices, such a property is necessary for a successful qualitative determination of any of the elements of \(A^{-1}\). New results presented include: (1) The basic necessary and sufficient characteristics of qualitatively invertible matrices given in a manner that consolidates earlier results and does not require transformation of the matrix (Theorem 1). (2) For A irreducible and qualitatively invertible, the nonzero signs in the transpose of A are also the signs of the corresponding elements in \(A^{-1}\) (Theorem 3). (3) For A irreducible and qualitatively invertible, the sign of an element in \(A^{-1}\) is not determinable iff the corresponding element in the transpose of A must be zero for A qualitatively invertible (Theorem 4).