Let T and S be n Times n upper triangular matrices Use the d

Let T and S be (n Times n) upper triangular matrices. Use the definition of matrix multiplication to show that the product ST is also upper triangular.

Solution

Given that T is upper triangular and S is upper triangular

By product of matrices defintion we multiply the first row with first column and add to get the element a11 of the product matrix.

Hence for a21, the product would be multiplication of second row of S with second column of T.

Second row of S have first element 0 and others non zero while T has in the second column first two non zero and others 0.

Hence product would result in only two non zero in the second row.

This will be true for any kth row of S and kth column of T.

Hence the product will also be upper triangular