Checkerboard inverse of a totally nonnegative matrix
= Checkerboard inverse of a totally nonnegative matrix
{title2=$(-1)^{i+j}(A^{-1})_{ij}\geq0$}
If an invertible square <totally nonnegative matrix> $A$ has size $n$, then $\det A>0$ and
$$
(-1)^{i+j}(A^{-1})_{ij}\geq0.
$$
Indeed the <adjugate identity> gives $(A^{-1})_{ij}=(-1)^{i+j}\det A_{\widehat j,\widehat i}/\det A$, and the deleted-row/deleted-column minor is nonnegative. Consequently an alternating vector $v_j=(-1)^j$ gives $(A^{-1}v)_i=(-1)^i\sum_j|(A^{-1})_{ij}|$.