Question

If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + B^T)^T = A^T + B.

Answer 1

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

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

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

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

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

∴ (A + B^T)^T = `[(1, -2),(-1, 4),(0, 0)]`       ...(i)

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

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

= `[(1, -2),(-1, 4),(0, 0)]`                 ...(ii)
From (i) and (ii), we get
(A + B^T)^T = A^T + B.