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प्रश्न
If A = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`, find A−1. Using A−1, solve the system of linear equations x − 2y = 10, 2x − y − z = 8, −2y + z = 7.
If A = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`, then find A−1.
Hence, solve the system of linear equations:
x − 2y = 10
2x − y − z = 8
−2y + z = 7
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उत्तर
A = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`
|A| = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`
= 1(−1 − 2) − 2(−2 + 0) + 0
= −3 + 4
= 1
|A| ≠ 0 A−1 exist.
Now find minors and cofactors
A11 = M11 = −3
A12 = −M12 = −(−2)
= 2
A13 = M13 = 2
A21 = −M21 = −2
A22 = M22 = 1
A23 = −M23 = −(−1)
= 1
A31 = M31 = −4
A32 = −M32 = (−2)
= −2
A33 = M33 = (−1 + 4)
= 3
adj A = [Cofactor matrix] = `[(-3, 2, 2), (-2, 1, 1), (-4, 2, 3)]`
= `[(-3, -2, -4), (2, 1, 2), (2, 1, 3)]`
`A^(-1)1/|A| adj A = 1/(+1)[(-3, -2, -4), (2, 1, 2), (2, 1, 3)]`
Given: x − 2y = 10
2x − y − z = 8
−2y + z = 7
In matrix form, `[(1, -2, 0), (2, -1, -1), (0, -2, 1)][(x), (y), (z)] = [(10), (8), (7)]`
A'X = B
X = (A')−1
B = (A−1)'B
`[(x), (y), (z)] = 1/1[(-3, -2, -4), (2, 1, 2), (2, 1, 3)][(10), (8), (7)]`
= `[(-3, 2, 2), (-2, 1, 1), (-4, 2, 3)][(10), (8), (7)]`
`[(x), (y), (z)] = [(-30 + 16 + 14), (-20 + 8 + 7), (-40 + 16 + 21)]`
= `[(-30 + 30), (-20 + 15), (-40 + 37)]`
`[(x), (y), (z)] = [(0), (-5), (-3)]`
Hence, x = 0, y = −5, z = −3.
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