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

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In this paper we construct analytic-numerical solutions for initial-boundary value systems related to the equation ${u}_{t}-A{u}_{xx}-Bu=0$, where $B$ is an arbitrary square complex matrix and $A$ ia s matrix such that the real part of the eigenvalues of the matrix $\frac{1}{2}(A+{A}^{H})$ is positive. Given an admissible error $\epsilon $ and a finite domain $G$, and analytic-numerical solution whose error is uniformly upper bounded by $\epsilon $ in $G$, is constructed.

Fiedler and Markham (1994) proved $${\left(\frac{\mathrm{det}\widehat{H}}{k}\right)}^{k}\ge \mathrm{det}H,$$ where $H={\left({H}_{ij}\right)}_{i,j=1}^{n}$ is a positive semidefinite matrix partitioned into $n\times n$ blocks with each block $k\times k$ and $\widehat{H}={\left(\mathrm{tr}{H}_{ij}\right)}_{i,j=1}^{n}$. We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $$\mathrm{det}({I}_{n}+\widehat{H})\ge \mathrm{det}{({I}_{nk}+kH)}^{1/k}.$$