Question

If A = `[(2, -3),(5, -4),(-6, 1)]`, B = `[(2, 1),(4, -1),(-3, 3)]` and 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)]`  ...(1)

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

From (1) and (2),

(A – C)^T = A^T – C^T