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Math Help - 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 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.
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