Question

Find matrices A and B, if `2"A" - "B" = [(6, -6, 0),(-4, 2, 1)] and "A" - 2"B" = [(3, 2, 8),(-2, 1, -7)]` 

Answer 1

Given equations are

2A – B =  `[(6, -6, 0),(-4, 2, 1)]`  ...(i)

 and A – 2B = `[(3, 2, 8),(-2, 1, -7)]`  ...(ii)

By (i) – (ii) × 2, we get

3B = `[(6, -6, 0),(-4, 2, 1)] -2[(3, 2, 8),(-2, 1, -7)]`

= `[(6, -6, 0),(-4, 2, 1)] - [(6, 4, 16),(-4, 2, -14)]`

= `[(6 - 6, -6 - 4, 0- 16),(-4 + 4, 2 - 2, 1 + 14)]`

∴ 3B = `[(0, -10, -16),(0, 0, 15)]`

∴ B = `1/3[(0, -10, -16),(0, 0, 15)]`

∴ B = `[(0, (-10)/3, (-16)/3),(0, 0, 5)]`

By (i) × 2 – (ii), we get

3A = `2[(6, -6, 0),(-4, 2, 1)] - [(3, 2, 8),(-2, 1, -7)]`

= `[(12, -12, 0),(-8, 4, 2)] - [(3, 2, 8),(-2, 1, -7)]`

= `[(12 - 3, -12 - 2, 0 - 8),(-8 + 2, 4 - 1, 2 + 7)]`

∴ 3A = `[(9, -14, -8),(-6, 3, 9)]`

∴ A = `1/3[(9, -14, -8),(-6, 3, 9)]`

∴ A = `[(3, (-14)/3, -8/3),(-2, 1, 3)]`