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Question
Find the inverse of the following matrix.
`[(0,1,2),(1,2,3),(3,1,1)]`
Solution
Let A = `[(0,1,2),(1,2,3),(3,1,1)]`
∴ |A| = `[(0,1,2),(1,2,3),(3,1,1)]`
= 0(2 − 3) − 1(1 − 9) + 2(1 − 6)
= 0 + 8 − 10
= −2 `≠` 0
∴ A−1 exists.
consider AA−1 = I
∴ `[(0,1,2),(1,2,3),(3,1,1)]`A−1 = `[(1,0,0),(0,1,0),(0,0,1)]`
By R1 ↔ R2, we get,
`[(1,2,3),(0,1,2),(3,1,1)]`A−1 = `[(0,1,0),(1,0,0),(0,0,1)]`
By R3 – 3R1, we get,
`[(1,2,3),(0,1,2),(0,-5,-8)]`A−1 = `[(0,1,0),(1,0,0),(0,-3,1)]`
By R1 – 2R2 and R3 + 5R2, we get,
`[(1,0,-1),(0,1,2),(0,0,2)]`A−1 = `[(-2,1,0),(1,0,0),(5,-3,1)]`
By `(1/2)`R3, we get,
`[(1,0,-1),(0,1,2),(0,0,1)]`A−1 = `[(-2,1,0),(1,0,0),(5//2,-3//2,1//2)]`
By R1 + R3 and R2 − 2R3, we get,
`[(1,0,0),(0,1,0),(0,0,1)]`A−1 = `[(1//2,-1//2,1//2),(-4,3,-1),(5//2,-3//2,1//2)]`
∴ A−1 = `1/2[(1,-1,1),(-8,6,-2),(5,-3,1)]`
[Note: Answer in the textbook is incorrect.]
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