Product_of_Triangular_Matrices

Let $\mathbf A = \sqbrk a_n, \mathbf B = \sqbrk b_n$ be upper triangular matrices of order $n$. Let $\mathbf C = \mathbf A \mathbf B$. Then $(1): \quad$ the diagonal elements of $\mathbf C$ are given by $\forall j \in \closedint 1 n: c_{j j} = a_{j j} b_{j j}$ That is, the diagonal elements of $\mathbf C$ are those of the factor matrices multiplied together. $(2): \quad$ The matrix $\mathbf C$ is itself upper triangular. The same applies if both $\mathbf A$ and $\mathbf B$ are lower triangular matrices.