Question

Answer the following question:

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

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

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

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

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

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

From (i) and (ii), we get

(3A – 5B^T)^T = 3A^T – 5B.