Question

Solve the following:

If A = `[(2, -3),(3, -2),(-1, 4)],"B" = [(-3, 4, 1),(2, -1, -3)]`, verify (3A – 5B^T)^T = 3A^T – 5B.

Answer 1

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

= `[(6, -),(9, -6),(-3, 12)] - [(-15, 10),(20, -5),(5, -15)]`

= `[(6 + 15, -9 - 10),(9 - 20, -6 + 5),(-3 - 5, 12 + 15)]`

∴ 3A – 5B^T = `[(21, -19),(-11, -1),(-8, 27)]`

∴ (3A – 5B^T)^T = `[(21, -11, -8),(-19, -1, 27)]`     ...(i)

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

= `[(6, 9, -3),(-9, 16, 12)] - [(-15, 20, 5),(10, -5, -15)]`

= `[(6 + 15, 9 - 20, -3 - 5),(-9 - 10, -6 + 5, 12 + 15)]`

= `[(21, -11, -8),(-19, -1, 27)]`      ...(ii)

From (i) and (ii), we get

(3A – 5B^T)^T = 3A^T – 5B