# Question_P2 (a)

The product of upper triangular matrices is upper triangular. We call a matrix A upper triangular if whenever i > j we have A_{ij} = 0. Let A be an m \times n upper triangular matrix and B be an n \times p upper triangular matrix. Then:

\displaystyle (AB)_{ij} = \sum_{k = 1}^n A_{ik} B_{kj}

Note that whenever i > k we have A_{ik} = 0, so the terms before k = i are zero meaning:

\displaystyle \sum_{k = 1}^n A_{ik} B_{kj} = \sum_{k = i}^n A_{ik} B_{kj}

Suppose i > j. Then for all k \ge i we have k > j so B_{kj} = 0. So:

\displaystyle (AB)_{ij} = 0

for i > j. So AB is upper triangular.

(b)

The product of lower triangular matrices is lower triangular. We call a matrix A lower triangular if whenever i < j we have A_{ij} = 0.

We have:

\displaystyle (AB)_{ij} = \sum_{k = 1}^n A_{ik} B_{kj}

Note that whenever i < k we have A_{ik} = 0, so the terms beyond k = i are 0 meaning:

\displaystyle (AB)_{ij} = \sum_{k = 1}^{i} A_{ik} B_{kj}

Suppose i < j. Then for k \le i we have k < j, so B_{kj} = 0 since B is lower triangular. So:

(AB)_{ij} = 0

whenever i < j, so AB is lower triangular. 