Question

Answer the following question:

If A = `[(2, -4),(3, -2),(0, 1)]`, B = `[(1, -1, 2),(-2, 1, 0)]`, show that (AB)^T = B^TA^T

Answer 1

AB = `[(2, -4),(3, -2),(0, 1)] [(1, -1, 2),(-2, 1, 0)]`

= `[(2 + 8, -2 - 4, 4 + 0),(3 + 4, -3 - 2, 6 - 0),(0 - 2, 0 + 1, 0 + 0)]`

= `[(10, -6, 4),(7, -5, 6),(-2, 1, 0)]`

∴ (AB)^T = `[(10, 7, -2),(-6, -5, 1),(4, 6, 0)]`  ...(i)

Now, A^T = `[(2, 3, 0),(-4, -2, 1)]` and B^T = `[(1, -2),(-1, 1),(2, 0)]`

∴ B^TA^T = `[(1, -2),(-1, 1),(2, 0)] [(2, 3, 0),(-4, -2, 1)]`

= `[(2 + 8, 3 + 4, 0 - 2),(-2 - 4, -3 - 2, 0 + 1),(4 - 0, 6 - 0, 0 + 0)]`

∴ B^TA^T = `[(10, 7, -2),(-6, -5, 1),(4, 6, 0)]`  ...(ii)

From (i) and (ii), we get

(AB)^T = B^TA^T