 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2021/3/a/solution

(A B)_{ij} = \sum_{k = 1}^{n} a_{ik} b_{kj}

. A nonzero summand in a product of upper triangular matrices requires both

i \leq k

and

k \leq j

. If

i > j

, no such index exists, so the product is upper triangular. If

i = j

, only

k = i

survives, giving

(A B)_{ii} = a_{ii} b_{ii} .

This also shows inductively why the diagonal of an integer matrix power consists of the corresponding powers of the diagonal entries.

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