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प्रश्न
Examine the consistency of the system of equations.
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
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उत्तर
Let, A = `[(3,-1,-2),(0,2,-1),(3,-5,0)]`, X = `[(x),(y),(z)]`, B = `[(2),(-1),(3)]`
|A| `= |(3,-1,-2),(0,2,-1),(3,-5,0)|`
= 3[2 × 0 + 5 × (−1)] + 1(0 + 3) − 2(0 − 6)
= −15 + 3 + 12
= 0
Cofactors of the elements of |A|:
A11 = `|(2,-1),(-5,0)|`
= 0 − 5
= −5
A12 = `-|(0,-1),(3,0)|`
= −(0 + 3)
= −3
A13 = `|(0,2),(3,-5)|`
= 0 − 6
= −6
A21 = `-|(-1,-2),(-5,0)|`
= −(0 − 10)
= 10
A22 = `|(3,-2),(3,0)|`
= 0 + 6
= 6
A23 = `-|(3,-1),(3,-5)|`
= −(−15 + 3)
= 12
A31 = `|(-1,-2),(2,-1)|`
= 1 + 4
= 5
A32 = `-|(3,-2),(0,-1)|`
= −(−3 + 0)
= 3
A33 = `|(3,-1),(0,2)|`
= 6 + 0
= 6
∴ |A| = `[(-5,-3,-6),(10,6,12),(5,3,6)]`
∴ (adj A) = `[(-5,10,5),(-3,6,3),(-6,12,6)]`
(adj A)B = ` [(-5,10,5),(-3,6,3),(-6,12,6)] [(2),(-1),(3)]`
= `[(-10-10 + 15),(-6 - 6 + 9),(-12 - 12 + 18)]`
= `[(-5),(-3),(-6)] ≠ 0`
∴ |A| = 0 and (adj A)B ≠ 0
Hence, the system of equations is inconsistent.
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