Question

Find matrices A and B, if 2A – B = `[(6, -6, 0),(-4, 2, 1)]` and A – 2B = `[(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) x 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) x 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)]`.