Question

If the system of equations lx₁ + x₂ + x₃ = 1, x₁ + lx₂ + x₃ = 1, x₁ + x₂ + lx₃ = 1 is consistent, then l can be-

• A. 5
• B. –2/3
• C. –3
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Let D = $\left| \begin{matrix} \lambda & 1 & 1 \\ 1 & \lambda & 1 \\ 1 & 1 & \lambda \end{matrix} \right|$ = $\left| \begin{matrix} \lambda + 2 & 1 & 1 \\ \lambda + 2 & \lambda & 1 \\ \lambda + 2 & 1 & \lambda \end{matrix} \right|$

[C₁ ® C₁ + C₂ + C₃]

= (l + 2) $\left| \begin{matrix} 1 & 1 & 1 \\ 1 & \lambda & 1 \\ 1 & 1 & \lambda \end{matrix} \right|$

= (l + 2) $\left| \begin{matrix} 1 & 0 & 0 \\ 1 & \lambda - 1 & 0 \\ 1 & 0 & \lambda - 1 \end{matrix} \right|$ = (l + 2) (l – 1)²

[using C₂ ® C₂ – C₁ and C₃ ® C₃ – C₁]

If D = 0, then l = –2 or l = 1. But when l = 1, the system of equation becomes x₁ + x₂ + x₃ = 1 which has infinite number of solutions. When l = – 2, by adding three equations, we obtain 0 = 3 and thus, the system of equations is inconsistent.