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Question
If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find A−1. Using A−1 solve the system of equations:
2x – 3y + 5z = 11
3x + 2y – 4z = –5
x + y – 2z = –3
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Solution
|A| = `|(2,-3,5),(3,2,-4),(1,1,-2)|`
= 2[2 × (−2) − 1 × (−4)] − (−3)[3 × (−2) − (1) × (−4)] + 5[3 × 1 − 1 × 2]
= 2[−4 + 4] + 3[−6 + 4] + 5[3 − 2]
= 0 + 3 × (−2) + 5 × 1
= −6 + 5
= −1 ≠ 0
∴ A−1 can be known,
Cofactors of the elements of |A|:
A11 = `|(2,-4),(1,-2)|`
= −4 + 4
= 0
A12 = `-|(3,-4),(1,-2)|`
= −(−6 + 4)
= 2
A13 = `|(3,2),(1,1)|`
= 3 − 2
= 1
A21 = `- |(-3,5),(1,-2)|`
= −(6 − 5)
= −1
A22 = `|(2,5),(1,-2)|`
= −4 − 5
= −9
A23 = `- |(2,-3),(1,1)|`
= −(2 + 3)
= −5
A31 = `|(-3,5),(2,-4)|`
= 12 − 10
= 2
A32 = `-|(2,5),(3,-4)|`
= −(−8 − 15)
= 23
A33 = `|(2,-3),(3,2)|`
= 4 + 9
= 13
The cofactor matrix of the elements of |A| = `[(0,2,1),(-1,-9,-5),(2,23,13)]`
∴ adj A = `[(0,2,1),(-1,-9,-5),(2,23,13)] = [(0,-1,2),(2,-9,23),(1,-5,13)]`
∴ A−1 = `1/|A|` adj A
= `1/(-1)[(0,-1,2),(2,-9,23),(1,-5,13)]`
= `[(0,1,-2),(-2,9,-23),(-1,5,-13)]`
Writing the given equation in the form AX = B,
Or A = `[(2,-3,5),(3,2,-4),(1,1,-2)]`, X = `[(x),(y),(z)]`, B = `[(11),(-5),(-3)]`
∴ X = A−1B
`[(x),(y),(z)] = [(0,1,-2),(-2,9,-23),(-1,5,-13)] [(11),(-5),(-3)]`
= `[(0 - 5 + 6),(-22 - 45 + 69),(-11 - 25 + 39)]`
= `[(1),(2),(3)]`
⇒ x = 1, y = 2, z = 3
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