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in Matrices & determinants by (41.3k points)

Show that the product of two triangular matrices is itself triangular.

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 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, cikaij 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.

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