Square Matrices- Upper Triangular

Quote:

Originally Posted by amm345

Show that if A and B are square matrices in the field F and are upper triangular, so is AB.

let $A=[a_{ij}], \ B=[b_{ij}].$ so we have $a_{ij}=b_{ij}=0$ whenever $1 \leq j < i \leq n.$ let $C=AB=[c_{ij}]$ then $c_{ij}=\sum_{k=1}^n a_{ik}b_{kj}.$ so if $j < i$ and $1 \leq k \leq n,$ then we cannot have $k \geq i$ and $k \leq j$ at the

same time. so for any $1 \leq k \leq n$ either $k < i$ or $k > j$ and thus either $a_{ik}=0$ or $b_{kj}=0.$ therefore if $j < i,$ then $a_{ik}b_{kj}=0$ for all $1 \leq k \leq n$ and so $c_{ij}=0.$ Q.E.D.