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Solve the following equation by the method of inversion: 2x - y = - 2, 3x + 4y = 3 - Mathematics and Statistics

Sum

Solve the following equation by the method of inversion:

2x - y = - 2, 3x + 4y = 3

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Solution

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.]

  Is there an error in this question or solution?
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