Chapter 1.6, Problem 27E

A square matrix $A={\left[a_{ij}\right]}_n$ with $a_{ij}=0$ for all $i>j$ is called upper triangular. Prove or disprove each of the following statements.

The set of all upper triangular matrices is closed with respect to matrix addition in $M_n\left(ℝ\right)$.

The set of all upper triangular matrices is closed with respect to matrix multiplication in $M_n\left(ℝ\right)$.

If $A$ and $B$ are square and the product $AB$ is upper triangular, then at least one of $A$ or $B$ is upper triangular.