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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)]`
Solution
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)]`
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