Question

If the system of simultaneous linear equations $x+\lambda y-2z=1$, $x-y+\lambda z=2$ and $x-2y+3z=3$ is inconsistent for $\lambda = \lambda_1$ and $\lambda_2$, then $\lambda_1 + \lambda_2 =$

• A. 5
• B. $\sqrt{5}$
• C. 1
• D. -1

Hint:

For a system of 3 linear equations, the condition for having no solution or infinitely many solutions is $\det(A) = 0$. The question asks for values where the system is inconsistent, which directly points to this condition. If the resulting equation in the parameter (here, $\lambda$) is quadratic, you can often find the sum or product of the parameter values using Vieta's formulas without explicitly solving for them.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
A system of non-homogeneous linear equations, represented as \( AX = B \), is inconsistent (has no solution) if the determinant of the coefficient matrix \( A \) is zero, and at least one of the determinants \( \Delta_x, \Delta_y, \Delta_z \) is non-zero. The primary condition to check is \( \det(A) = 0 \). 
Step 2: Key Formula or Approach 
The given system of equations is: \[ x + \lambda y - 2z = 1 \] \[ x - y + \lambda z = 2 \] \[ x - 2y + 3z = 3 \] We first write the coefficient matrix \( A \) and set its determinant to zero to find the values of \( \lambda \) for which the system does not have a unique solution. 
Step 3: Detailed Explanation 
The coefficient matrix \( A \) is:
\[ A = \begin{pmatrix} 1 & \lambda & -2 \\ 1 & -1 & \lambda \\ 1 & -2 & 3 \end{pmatrix} \] For the system to be inconsistent, we must have \( \det(A) = 0 \).
\[ \det(A) = 1 \begin{vmatrix} -1 & \lambda \\ -2 & 3 \end{vmatrix} - \lambda \begin{vmatrix} 1 & \lambda \\ 1 & 3 \end{vmatrix} + (-2) \begin{vmatrix} 1 & -1 \\ 1 & -2 \end{vmatrix} = 0 \] \[ 1((-1)(3) - (\lambda)(-2)) - \lambda((1)(3) - (\lambda)(1)) - 2((1)(-2) - (-1)(1)) = 0 \] \[ 1(-3 + 2\lambda) - \lambda(3 - \lambda) - 2(-2 + 1) = 0 \] \[ -3 + 2\lambda - 3\lambda + \lambda^2 - 2(-1) = 0 \] \[ \lambda^2 - \lambda - 3 + 2 = 0 \] \[ \lambda^2 - \lambda - 1 = 0 \] This is a quadratic equation in \( \lambda \). The problem states that the system is inconsistent for two values, \( \lambda_1 \) and \( \lambda_2 \). These must be the roots of this quadratic equation. 
(We should verify that for these roots, the system is indeed inconsistent and not consistent with infinite solutions. This requires checking that \( (\text{adj } A)B \neq 0 \), or that at least one of \( \Delta_x, \Delta_y, \Delta_z \) is non-zero. For the roots of \( \lambda^2 - \lambda - 1 = 0 \), this condition holds, leading to inconsistency). 
According to Vieta's formulas for a quadratic equation \( ax^2+bx+c=0 \), the sum of the roots is \( -b/a \). 
For the equation \( \lambda^2 - \lambda - 1 = 0 \), the sum of the roots \( \lambda_1 + \lambda_2 \) is:
\[ \lambda_1 + \lambda_2 = - \frac{-1}{1} = 1 \] Step 4: Final Answer 
The sum of the values of \( \lambda \) for which the system is inconsistent is \( \lambda_1 + \lambda_2 = 1 \).