Sum of Triangular Matrices

Theorem

Let $\mathbf A = \left[{a}\right]_{n}, \mathbf B = \left[{b}\right]_{n}$ be square matrices of order $n$.

Let $\mathbf C = \mathbf A + \mathbf B$ be the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

If $\mathbf A$ and $\mathbf B$ are upper triangular matrices, then so is $\mathbf C$.

If $\mathbf A$ and $\mathbf B$ are lower triangular matrices, then so is $\mathbf C$.

Proof

From the definition of matrix addition, we have:

$\forall i, j \in \left[{1 .. n}\right]: c_{ij} = a_{ij} + b_{ij}$

If $\mathbf A$ and $\mathbf B$ are upper triangular matrices, we have:

$\forall i > j: a_{ij} = b_{ij} = 0$

Hence:

$\forall i > j: c_{ij} = a_{ij} + b_{ij} = 0 + 0 = 0$

and so $\mathbf C$ is itself upper triangular.

Similarly, if $\mathbf A$ and $\mathbf B$ are lower triangular matrices, we have:

$\forall i < j: a_{ij} = b_{ij} = 0$

Hence:

$\forall i < j: c_{ij} = a_{ij} + b_{ij} = 0 + 0 = 0$

and so $\mathbf C$ is itself lower triangular.

$\blacksquare$