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Solve the following equations by inversion method. x + 2y = 2, 2x + 3y = 3 - Mathematics and Statistics

Sum

Solve the following equations by inversion method.

x + 2y = 2, 2x + 3y = 3

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Solution

The given equations can be written in the matrix form as:

`[(1,2),(2,3)][("x"),("y")]=[(2),(3)]`

This is of the form AXB, where

A = `[(1,2),(2,3)]`, x = `[("x"),("y")]` and B = `[(2),(3)]`

Let us find A−1

|A| = `[(1,2),(2,3)]` = 3 − 4 = − 1 `≠` 0

∴ A−1 exists.

Consider AA−1 = I

∴ `[(1,2),(2,3)]`A−1 = `[(1,0),(0,1)]`

Bu R2 − 2R1, we get,

`[(1,2),(0,-1)]`A−1 = `[(1,0),(-2,1)]`

By (− 1)R2, we get,

`[(1,2),(0,1)]`A−1 = `[(1,0),(2,-1)]`

By R1 − 2R2, we get,

`[(1,0),(0,1)]`A−1 = `[(-3,2),(2,-1)]`

∴ A−1 = `[(-3,2),(2,-1)]`

Now, premultiply AX = B by A−1, we get,

A−1(AX) = A−1B

∴ (A−1A)X = A−1B

∴ IX = A−1B

∴ X = `[(-3,2),(2,-1)][(2),(3)]`

∴ `[("x"),("y")]` = `[(-6+6),(4-3)]` = `[(0),(1)]`

By equality of matrices,

x = 0, y = 1 is the required solution.

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