### A 1-norm bound for inverses of triangular matrices with monotone entries.

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In this paperwe study the Hadamard product of inverse-positive matrices.We observe that this class of matrices is not closed under the Hadamard product, but we show that for a particular sign pattern of the inverse-positive matrices A and B, the Hadamard product A ◦ B−1 is again an inverse-positive matrix.

A simple proof is given for a theorem by Milliken and Akdeniz (1977) about the difference of the Moore-Penrose inverses of two positive semi-definite matrices.

It is shown that $$\text{rank}\left({P}^{*}AQ\right)=\text{rank}\left({P}^{*}A\right)+\text{rank}\left(AQ\right)-\text{rank}\left(A\right),$$ where $A$ is idempotent, $[P,Q]$ has full row rank and ${P}^{*}Q=0$. Some applications of the rank formula to generalized inverses of matrices are also presented.

It is proved in this paper that special generalized ultrametric and special $\mathcal{U}$ matrices are, in a sense, extremal matrices in the boundary of the set of generalized ultrametric and $\mathcal{U}$ matrices, respectively. Moreover, we present a new class of inverse $M$-matrices which generalizes the class of $\mathcal{U}$ matrices.