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Question
If A and B are square matrices of the same order, then (AB)′ = ______.
Solution
If A and B are square matrices of the same order, then (AB)′ = B'A'.
Explanation:
Let A = [aij]m × n and B = [bij]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"`
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