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Thread: Square Matrices- Upper Triangular

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    Square Matrices- Upper Triangular

    Show that if A and B are square matrices in the field F and are upper triangular, so is AB.
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    Quote Originally Posted by amm345 View Post
    Show that if A and B are square matrices in the field F and are upper triangular, so is AB.
    let $\displaystyle A=[a_{ij}], \ B=[b_{ij}].$ so we have $\displaystyle a_{ij}=b_{ij}=0$ whenever $\displaystyle 1 \leq j < i \leq n.$ let $\displaystyle C=AB=[c_{ij}]$ then $\displaystyle c_{ij}=\sum_{k=1}^n a_{ik}b_{kj}.$ so if $\displaystyle j < i$ and $\displaystyle 1 \leq k \leq n,$ then we cannot have $\displaystyle k \geq i$ and $\displaystyle k \leq j$ at the

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