The M-matrix inverse problem for singular and symmetric Jacobi matrices

We aim here at characterizing when the Moore-Penrose inverse of a singular and symmetric Jacobi M-matrix is also an M-matrix. This characterization involves a highly non-linear system of inequalities in the off-diagonal entries of the matrix. We obtain all the solutions of this system for n⩽3 but when n⩾4, the system becomes much more complicated. Our main result establishes that for any n, there exist singular, symmetric and tridiagonal M-matrices of order n whose Moore-Penrose inverse is also an M-matrix.