Question

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

Answer 1

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

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

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

= `[(0 - 2, 6 + 1, -8 - 1),(0 - 4, 9 + 2, -12 - 2),(0 + 2, 12 - 1, -16 + 1)]`

= `[(-2, 7, -9),(-4, 11, -14),(2, 11, -15)]`

∴ (AB)^T = `[(-2, -4, 2),(7, 11, 11),(-9, -14, -15)]`     ...(i)

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

= `[(0 - 2, 0 - 4, 0 + 2),(6 + 1, 9 + 2, 12 - 1),(-8 - 1, -12 - 2, -16 + 1)]`

= `[(-2, -4, 2),(7, 11, 11),(-9, -14, -15)]`          ...(ii)
From (i) and (ii), we get
(AB)^T = B^TA^T.