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प्रश्न
Find the inverse of the matrices (if it exists).
`[(1,-1,2),(0,2,-3),(3,-2,4)]`
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उत्तर
Let A = `[(1,-1,2),(0,2,-3),(3,-2,4)]`
∴ |A| = `|(1,-1,2),(0,2,-3),(3,-2,4)|`
= 1(8 − 6) + 1(0 + 9) + 2(0 − 6)
= 2 + 9 − 12
= −1 ≠ 0
∴ A is invertible.
Let Cij be the cofactor of aij in A; then the cofactors of elements of A are given by,
C11 = `(-1)^(1+1) |(2,-3), (-2,4)|`
= 8 − 6
= 2
C12 = `(-1)^(1+2) |(0,-3), (3,4)|`
= −(0 + 9)
= −9
C13 = `(-1)^(1+3)|(0,2),(3,-2)|`
= 0 − 6
= −6
C21 = `(-1)^(2+1) |(-1,2), (-2,4)|`
= −(−4 + 4)
= 0
C22 = `(-1)^(2+2) |(1,2), (3,4)|`
= 4 − 6
= −2
C23 = `(-1)^(2+3) |(1,-1), (3,-2)|`
= −(−2 + 3)
= −1
C31 = `(-1)^(3+1) |(-1,2), (2,-3)|`
= 3 − 4
= −1
C32 = `(-1)^(3+2) |(1,2), (0,-3)|`
= −(−3 − 0)
= 3
C33 = `(-1)^(3+3)|(1,-1), (0,2)|`
= 2 + 0
= 2
∴ adj A = `[(2,-9,-6),(0,-2,-1),(-1,3,2)] = [(2,0,-1),(-9,-2,3),(-6,-1,2)]`
A−1 = `1/|A|` adj A
= `1/-1[(2,0,-1),(-9,-2,3),(-6,-1,2)]`
= `[(-2,0,1),(9,2,-3),(6,1,-2)]`
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