Question

Solve the system of linear equations using the matrix method.

2x – y = –2

3x + 4y = 3

Answer 1

Given system of equations,

2x – y = –2
3x + 4y = 3

The system of equations can be written as AX = B. Hence, X = A⁻¹B.

A = `[(2,-1),(3,4)]`, X = `[(x),(y)]` and B = `[(-2),(3)]`

⇒ |A| = `|(2,-1),(3,4)|`

= 8 + 3

= 11 ≠ 0

The cofactors of the elements of matrix A are as follows:

A₁₁ = (−1)¹⁺¹ (4) = 4

A₁₂ = (−1)¹⁺² (3) = −3

A₂₁ = (−1)²⁺¹ (−1) = 1

A₂₂ = (−1)²⁺² (2) = 2

Matrix made of elements of cofactor of A = `[(4,-3),(1,2)]`

adj A = `[(4,-3),(1,2)] = [(4,1),(-3,2)]`

A⁻¹ = `1/|A|` (adj A)

= `1/11 [(4,1),(-3,2)]`

∴ X = A⁻¹B

= `1/11 [(4,1),(-3,2)] [(-2),(3)]`

= `1/11 [(-8 + 3),(6 + 6)]`

= `1/11 [(-5),(12)]`

= `[(-5/11),(12/11)]`

⇒ `[(x),(y)] = [(-5/11),(12/11)]`

⇒ x = `-5/11` and y = `12/11`