For each efficient vector w associated with an n-by-n reciprocal matrix A, we may form the matrix \([\frac{w_{i}}{w_{j}}].\) Our primary purpose is to give tight lower and upper bound matrices \(L_{A}\) and \(U_{A}\) such that \(L_{A}\le \) \([\frac{w_{i}}{w_{j}}]\le U_{A}\) (entry-wise). These matrices are calculated from the entries of A and we give insights into their special structure. We then use \(L_{A}\) and \(U_{A}\) to learn more about the cardinal ranking of alternatives associated with the pair-wise ratio comparisons in A. This includes (1) analysis of rank reversals; (2) a transparent characterization of the reciprocal matrices whose efficient vectors have a uniform order; and (3) a simple scheme to select a "best order" on the underlying alternatives. We discuss when two reciprocal matrices with the same set of efficient vectors are the same. We also mention new possible indices of inconsistency based upon \(L_{A}\) , \(U_{A}\) . They are compared to the classical measure \((\lambda (A)-n)/(n-1)\) , in which \(\lambda (A)\) is the Perron root of A.
