On a conjecture concerning the extensions of a reciprocal matrix

Let A be a reciprocal matrix of order n and w be its Perron eigenvector. To infer the efficiency of w for A, based on the principle of Pareto optimal decisions, we study the strong connectivity of a certain digraph associated with A and w. A reciprocal matrix B of order $n+1$ is an extension of A if the matrix A is obtained from B by removing its last row and column. We prove that there is no extension of a reciprocal matrix whose digraph associated with the extension and its Perron eigenvector has a source, as conjectured by Furtado and Johnson in ``Efficiency analysis for the Perron vector of a reciprocal matrix". As an application, considering n≥ 5 and A a matrix obtained from a consistent one by perturbing four entries above the main diagonal, $x,y,z,a$, and the corresponding reciprocal entries, in a way that there is a submatrix of size $2$ containing the four perturbed entries and not containing a diagonal entry, we describe the relations among $x,y,z,a$ with which A always has efficient Perron eigenvector.