Question

If A and B are square matrices of the same order, then (AB)′ = ______.

Answer 1

If A and B are square matrices of the same order, then (AB)′ = B'A'.

Explanation:

Let A = [a_ij]_{m × n} and B = [b_ij]_{n × p} be two martices.

Then, AB is an m × p matrix.

Therefore (AB)' is a p × m matrix.

Since A' and B' are n × m and p ×n matrices.

Therefore B'A' is a p × m matrix.

Thus, the two matrices (AB)' and B'A' are of the same order such that ((AB)')_ij = (AB)_ij

= `sum_("r" = 1)^"n" "a"_"jr""b"_"ri"`

= `sum_("r" = 1)^"n" "b"_"ri""a"_"jr"`

= `sum_("r" = 1)^"n" ("B'")_"ir"("A'")_"rj"`

= `("B'A'")_"ij"`