Question

If the system of equations x + λy + 2 = 0, λx + y – 2 = 0, λx + λy + 3 = 0 is consistent, then

Options

• λ = ±1 

• λ = ±2 

• λ = 1, – 2

• λ = –1, 2

Answer 1

λ = ±1 

Explanation:

The system of equations will be consistent if 

Δ = `|(1, lambda, 2),(lambda, 1, -2),(lambda, lambda, 3)|` = 0

To evaluate Δ we use R₁ `→` R₁ + R₂ followed by C₂ `→` C₂ – C₁ to obtain

Δ = `|(lambda + 1, lambda + 1, 0),(lambda, 1, -2),(lambda, lambda, 3)| = |(lambda + 1, 0, 0),(lambda, 1 - lambda, -2),(lambda, 0, 3)|`

= `3(lambda + 1)(1 - lambda) = 3(1 - lambda^2)`

For the system to be consistent, we must have

`1 - lambda^2` = 0 or `lambda` = ±1