Question

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Answer 1

Let, A = `[(1,3),(2,6)]`, X = `[(x),(y)]`, B = `[(5),(8)]`

Then the given system of equations can be written as,

`[(1,3),(2,6)][(x),(y)] = [(5),(8)]`

Now, |A|

= `|(1,3),(2,6)|`

= 1 × 6 − 2 × 3

= 6 − 6

= 0

Cofactors of the elements of |A| are, respectively,

A₁₁ = 6, A₁₂ = −2, A₂₁ = −3, A₂₂ = 1

∴ adj A = `[(6,-2),(-3,1)] = [(6,-3),(-2,1)]`

⇒ (adj A)B = `[(6,-3),(-2,1)] [(5),(8)]`

= `[(30 - 24),(-10 + 8)]`

= `[(6),(-2)] ≠ 0`

|A| = 0 and (adj A)B ≠ 0

Therefore, the given system of equations is inconsistent.