For what value of α ∈ R the system of equation: x + y + z = 3, 2x + 4y + 2z = 6 and α x + α y + αz = 3α  has infinitely many solutions.

Concept:

Let us consider a system of equations in three variables:

a₁ × x + b₁ × y + c₁ × z = d₁

a₂ × x + b₂ × y + c₂ × z = d₂

a₃ × x + b₃ × y + c₃ × z = d₃

Then, 

By cramer’s rule:

I. If Δ ≠ 0, then the system of equation has unique solution and it is given by: 

II. If Δ = 0 and atleast one of the determinants Δ, Δ₁, Δ₂ and Δ₃ is non-zero, then the given system is inconsistent.

III. If Δ = 0 and Δ₁ =  Δ₂ =  Δ₃ = 0, then the system is consistent and has infinitely many solutions.

Calculation:

Given: x + y + z = 3, 2x + 2y + 2z = 6 and α x + α y + αz = 3α

As we know that,

⇒ Δ = 0 and Δ₁ =  Δ₂ =  Δ₃ = 0

Hence, the given system has infinite solution for α ∈ R.