Question

Solve the following equation by the method of inversion:

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

Answer 1

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.