Question

If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(2, 1),(4, -1),(-3, 3)], "C" = [(1, 2),(-1, 4),(-2, 3)]`, then show that (A – C)^T = A^T – C^T.

Answer 1

A – C = `[(2, -3),(5, -4),(-6, 1)] - [(1, 2),(-1, 4),(-2, 3)]`

= `[(2 - 1, -3 - 2),(5 + 1, -4 - 4),(-6 + 2, 1 - 3)]`

= `[(1, -5),(6, -8),(-4, -2)]`

∴ (A – C)^T = `[(1, 6, -4),(-5, -8, -2)]`              ...(i)

Now, A^T = `[(2, 5, -6),(-3, -4, 1)]` and

C^T = `[(1, -1, -2),(2, 4, 3)]`

∴ A^T –  C^T = `[(2, 5, -6),(-3, -4, 1)] - [(1, -1, -2),(2, 4, 3)]`

= `[(2 - 1, 5 + 1, -6 + 2),(-3 - 2, -4 - 4, 1 - 3)]`

= `[(1, 6, -4),(-5, -8, -2)]`          ...(ii)

From (i) and (ii), we get
(A – C)^T = A^T – C^T.