Question

Solve the following:

If A = `[(2, -3),(3, -2),(-1, 4)],"B" = [(-3, 4, 1),(2, -1, -3)]`, verify (A + 2B^T)^T = A^T + 2B.

Answer 1

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

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

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

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

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

∴ A + 2B^T = `[(-4, 1),(11, -4),(1, -2)]`

∴ (A + 2B^T)^T = `[(-4, 11, 1),(1, -4, -2)]`     ...(i)

A + 2B = `[(2, 3, -1),(-3, -2, 4)] + 2[(-3, 4, 1),(2, -1, -3)]`

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

= `[(-4, 11, 1),(1, -4, -2)]`      ...(ii)

From (i) and (ii), we get

(A + 2B^T)^T = A^T + 2B