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

Answer 1

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

∴ AT + BT = `[(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.