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Question
Determine the matrices A and B when A + 2B = `[(1, 2),(6, -3)] and 2"A" - "B" = [(2, -1),(2, -1)]`
Solution
A + 2B = `[(1, 2),(6, -3)]` ......(i)
2A – B = `[(2, -1),(2, -1)]` ......(ii)
Multiplying (i) by 1 and (ii) by 2
A + 2B = `[(1, 2),(6, -3)]`
4A – 2B = `2[(2, -1),(2, -1)] = [(4, -2),(4, -2)]`
Adding, we get
5A = `[(1, 2),(6, -3)] + [(4, -2),(4, -2)] = [(5, 0),(10, -5)]`
A = `(1)/(5)[(5, 0),(10, 5)] = [(1, 0),(2, -1)]`
From (i) A + 2B = `[(1, 2),(6, -3)]`
= `[(1, 0),(2, -1)] + 2"B" = [(1, 2),(6, -3)]`
2B = `[(1, 2),(6, -3)] - [(1, 0),(2, -1)] = [(0, 2),(4, -2)]`
∴ B = `(1)/(2)[(0, 2),(4, -2)] = [(0, 1),(2, -1)]`
Hence A = `[(1, 0),(2, -1)]and "B" = [(0, 1),(2, -1)]`.
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