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Question
If A = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`, then find A2 and hence find A−1
Solution
|A| = `0 - 4|(1, -3),(-1, 4)| + 3|(1, -3),(-1, 4)|`
= – 4(4 – 3) + 3(4 – 3)
= –1 ≠ 0
∴ A−1 exist
A2 = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)] [(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`
= `[(0 + 4 - 3, 0 - 12 + 12, 0 - 12 + 12),(0 - 3 + 3, 4 + 9 - 12, 3 + 9 - 12),(0 + 4 - 4, -4 - 12 + 16, -3 - 12 + 16)]`
= `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
i.e., A × A = I
∴ A–1 × A × A = A–1 × I
∴ A = A–1
∴ A–1 = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`
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