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प्रश्न
Solve the following equation by the method of inversion:
2x - y = - 2, 3x + 4y = 3
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उत्तर
The given equations can be written in the matrix form as:
`[(2,-1),(3,4)][("x"),("y")] = [(-2),(3)]`
This is of the form AX = B, where
A = `[(2,-1),(3,4)], "X" = [("x"),("y")] "and" "B" = [(-2),(3)]`
Let us find A-1.
|A| = `|(2,-1),(3,4)| = 8 + 3 = 11 ne 0`
∴ A-1 exists.
Consider AA-1 = I
∴ `[(2,-1),(3,4)] "A"^-1 = [(1,0),(0,1)]`
By R1 ↔ R2 we get,
`[(3,4),(2,-1)] "A"^-1 = [(0,1),(1,0)]`
By R1 - R2, we get,
`[(1,5),(2,-1)] "A"^-1 = [(-1,1),(1,0)]`
By R2 - 2R1, we get,
`[(1,5),(0,-11)] "A"^-1 = [(-1,1),(3,-2)]`
By `(- 1/11)"R"_2`, we get,
`[(1,5),(0,1)] "A"^-1 = [(-1,1),(-3/11,2/11)]`
By R1 - 5R2 we get
`[(1,0),(0,1)] "A"^-1 = [(4/11,1/11),(-3/11,2/11)]`
∴ A-1 = `[(4/11,1/11),(-3/11,2/11)]`
Now, premultiply AX = B by A-1 , we get,
A-1 (AX) = A-1 B
∴ (A-1 A)X = A-1 B
∴ IX = A-1 B
∴ X = `[(4/11,1/11),(-3/11,2/11)] [(-2),(3)]`
∴ `[("x"),("y")] = [(-8/11 + 3/11),(6/11 + 6/11)] = [(-5/11),(12/11)]`
By equality of matrices,
x = `- 5/11`, y = `12/11` is the required solution.
Notes
[Note: Question in the textbook is incomplete.]
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