Question

If A = `[(5, 2, 9),(1, 2, 8)]`, B = `[(1, 7),(1, 2),(5, -1)]` verify that (AB)^T = B^T A^T

Answer 1

Given A = `[(5, 2, 9),(1, 2, 8)]`, B = `[(1, 7),(1, 2),(5, -1)]`

AB = `[(5, 2, 9),(1, 2, 8)] xx [(1, 7),(1, 2),(5, -1)]`

= `[(5 + 2 + 45, 35 + 4 - 9),(1 + 2 + 40, 7 + 4 - 8)]`

= `[(52, 30), (43, 3)]`

(AB)^T = `[(52, 43), (30, 3)]`   ...(1)

B^T = `[(1, 1, 5),(7, 2, -1)]`

A^T =  `[(5, 1),(2, 2),(9, 8)]`

B^T A^T = `[(1, 1, 5),(7, 2, -1)] xx [(5, 1),(2, 2),(9, 8)]`

= `[(5 + 2 + 45, 1 + 2 + 40),(35 + 4 - 9, 7 + 4 - 8)]`

= `[(52, 43),(30, 3)]`   ...(2)

From (1) and (2) we get, (AB)^T = B^T A^T