Question

Solve the following equations by inversion method:

x + y = 4, 2x - y = 5

Answer 1

x + y = 4,  2x - y = 5

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

`[(1,1),(2,-1)] [("x"),("y")] = [(4),(5)]`

This is of the form AX = B,

Where A = `[(1,1),(2,-1)]`, X = `[("x"),("y")]` and B = `[(4),(5)]`

To determine X, we have to find A⁻¹.

|A| = `|(1,1),(2,-1)|` = −1 − 2 = −3 ≠ 0

∴ A⁻¹ exists.

Consider AA⁻¹ = I

∴ `[(1,1),(2,-1)] "A"^(-1) = [(1,0),(0,1)]`

Applying R₂ `->` R₂ − 2R₁, we get

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

Applying `"R"_2 -> |-1/3|"R"_2`, we get

`[(1,1),(0,1)] "A"^(-1) = [(1,0),(2/3,-1/3)]`

Applying R₁ `->` R₁ − R₂, we get

`[(1,0),(0,1)] "A"^(-1) = [(1/3,1/3),(2/3,-1/3)]`

∴ A⁻¹ = `1/3 [(1,1),(2,-1)]`

Pre-multiplying AX = B by A⁻¹, we get

A⁻¹(AX) = A⁻¹B

∴ (A⁻¹A)X = A⁻¹B

∴ IX = A⁻¹B

∴ X = A⁻¹B

∴ X = `1/3[(1,1),(2,-1)] = [(4),(5)]`

∴ `[("x"),("y")] = 1/3[(4 + 5),(8 - 5)]`

= `1/3[(9),(3)]`

= `[(3),(1)]`

∴ By equality of matrices, we get

x = 3, y = 1