# 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

#### 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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