Let A = [aij]n×n and B = [bjk]n×n be two triangular matrices.
Then aij = 0 when i > j.
Also bjk = 0 when j > k
Let AB = [cik]n×n.Then, cik = ∑aij bjk for j ∈ [j=1, n].
Suppose that i > k:
(1) If j < i, then aij = 0 and therefore cik = 0.
(2) If i < j, then j > k because i > k. In this case, bjk = 0.
and therefore cik = 0.
Thus, cik = 0 whenever i > k.
Hence, the matrix AB is also a triangular matrix.