Question

Solve the following equations by method of inversion.
x + 2y = 2, 2x + 3y = 3

Answer 1

Matrix form of the given system of equations is
`[(1, 2),(2, 3)] [(x),(y)] = [(2),(3)]`
This is of the form AX = B.
where A = `[(1, 2),(2, 3)], "X" = [(x),(y)] "and B" = [(2),(3)]`
To determine X, we have to 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)]`

Applying R₂ → R₂ – 2R₁, we get

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

Applying R₂ → (– 1)R₂, we get

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

Applying R₁ → R₁ – 2R₂, we get

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

∴ A^–1 = `[(-3, 2),(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 = `[(-3, 2),(2, -1)] [(2),(3)]`

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

∴ By equality of matrices, we get
x = 0 and y = 1.