A problem involving induction and upper triangular matrix

Question

Use induction to prove that, if A1, ..., An are upper triangular matrices of the same size, then $\sum_{i=1}^n A_i$ is upper triangular.

How do i do this problem?

Thank You

Answer 1

$A$ is upper triangular iff $e_i^T Ae_j = 0$ for all $i>j$.

Suppose $A,B$ are upper triangular, then $e_i^T (A+B)e_j = 0$ for all $i>j$ and so $A+B$ is upper triangular.

Hence if $A_1+\cdots +A_n$ and $A_{n+1}$ are upper triangular, then so is $A_1+\cdots +A_{n+1}$.