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.