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 + B)^T = A^T + B^T

Answer 1

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

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

= `[(4, -2),(9, -5),(-9, 4)]`

∴ (A + B)^T = `[(4, 9, -9),(-2, -5, 4)]`   ...(i)

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

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

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

= `[(4, 9, -9),(-2, -5, 4)]`   ...(ii)

From (i) and (ii), we get

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