The authors consider the sign inheritance problem for the \(LU\) factorization of \(P\)-matrices, that is, \(\text{sgn}(u_{ij})= \text{sgn}(a_{ij})\) for a given pair \(i\leq j\), or for all such pairs. An \(n\times n\) array \(B\) is a sign pattern (matrix) if each entry of \(B\) is \(+\), \(-\) or 0. A matrix \(A\) has the sign pattern of \(B\) if for all \(i\), \(j\) \(\text{sgn}(a_{ij})\) and the \((i, j)\) entry of \(B\) are the same. For a fixed sign pattern \(B\), \(P_ B\) denotes the set of all \(P\)- matrices with the sign pattern of \(B\). Let \(LU= A\in P_ B\). If for every \(A\in P_ B\), \(\text{sgn}(u_{ij})\) is uniquely one of \(+\), \(-\), 0 respectively, then \(u_{ij}\) is said to be unambiguous.
The authors determine for an \(n\times n\) \(P\)-matrix \(A= LU\) combinatorial circumstances for which \(u_{ij}\) is unambiguously signed for a given pair \(i\leq j\), or all such pairs. For sign nonsingular matrices with all diagonal entries positive, sufficient conditions for an entry \(u_{ij}\) or the matrix \(U\) to be unambiguous are presented. It is also proved that if \(A\) is a qualitative \(P\)-matrix with \(A^{- 1}\) unambiguously signed, then \(L\), \(U\), \(L^{- 1}\) and \(U^{- 1}\) are all unambiguous.