Question

7. Consider the Theorem: Suppose A and B are two lower triangular matrices (Defined in 8 3.1), of order n. Then, the product AB is also a lower triangular matrix. Likewise for upper triangular matrices. (We say that the set of lower triangular matrices, of order n, is closed under multiplication.) Prove this theorem, for n = 3, by multiplying the following two matri- ces: a1 0 0 A bi b 0 1 0 0 and B 2 0 21 22 23 Answer: AB-

Answer 1

A = (a_1 b_1 c_1 0 b_2 c_2 0 0 c_3) and B = [x_1 y_1 z_1 0 y_2 z_2 0 0 z_3] Both are Lower triangular matrix, Now order to find A B we have AB = [a_1 b_1 0 b_2 c_2 0 0 c_3] [x_1 y_1 z_1 0 y_2 z_2 0 0 z_3] [a_1 x_1 + 0 + 0 b_1 x_1 + b_2 y_1 + 0 c_1 x_1 + c_2 y_1 + c_3 z_1 0 + 0 + 0 0 + b_2 y_2 + 0 0 + c_2 y_2 + c_3 z_2 0 + 0 + 0 0 + 0 + 0 0 + 0 + c_3 z_3] (a_1 x_1 b_1 x_1 + b_2 y_1 c_1 x_1 + c_2 y_2 + c_3 z_1 0 b_2 y_2 c_2 y_2 + c_3 z_3 0 0 c_3 - z_3)_3 times 3 which is the Lower triangular matrix