Proof of Theorem PTMT

We prove this for lower triangular matrices and leave the proof for upper triangular matrices to you. Suppose that $A$ and $B$ are both lower triangular. We need only establish that certain entries of the product $AB$ are zero. Suppose that $i<j$, then \begin{align*} \matrixentry{AB}{ij} &= \sum_{k=1}^{n}\matrixentry{A}{ik}\matrixentry{B}{kj} &&&&\knowl{./knowls/theorem.EMP.knowl}{\text{Theorem EMP}}\\ &= \sum_{k=1}^{j-1}\matrixentry{A}{ik}\matrixentry{B}{kj} + \sum_{k=j}^{n}\matrixentry{A}{ik}\matrixentry{B}{kj} &&&&\knowl{./knowls/property.AACN.knowl}{\text{Property AACN}}\\ &= \sum_{k=1}^{j-1}\matrixentry{A}{ik}0 + \sum_{k=j}^{n}\matrixentry{A}{ik}\matrixentry{B}{kj} &k&<j\text{,}&&\knowl{./knowls/definition.LTM.knowl}{\text{Definition LTM}}\\ &= \sum_{k=1}^{j-1}\matrixentry{A}{ik}0 + \sum_{k=j}^{n}0\matrixentry{B}{kj} &i&<j\leq k\text{,}&&\knowl{./knowls/definition.LTM.knowl}{\text{Definition LTM}}\\ &= \sum_{k=1}^{j-1}0+\sum_{k=j}^{n}0\\ &= 0 \end{align*}

Since $\matrixentry{AB}{ij}=0$ whenever $i<j$, by Definition LTM, $AB$ is lower triangular.