The following system of equations:
\(x + y - z = 7\)
\(4x + \lambda y - \lambda z = 3\)
\(3x + 2y - 4z = 5\)
does not possess a solution if the value of \(\lambda\) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4
If the coefficient determinant is non-zero, the system has a unique solution. So a system with no solution needs
\( |A| = \begin{vmatrix}1&1&-1\\4&\lambda&-\lambda\\3&2&-4\end{vmatrix} = 0 \)
Expanding along the first row:
\( |A| = 1(-4\lambda + 2\lambda) - 1(-16 + 3\lambda) + (-1)(8 - 3\lambda) \)
\( = -2\lambda + 16 - 3\lambda - 8 + 3\lambda \)
\( = 8 - 2\lambda \)
So \(|A| = 0\) only for \(\lambda = 4\); for every other value the solution is unique. With \(\lambda = 4\) the second equation becomes
\( 4(x + y - z) = 3 \)
\( \displaystyle x + y - z = \frac{3}{4} \)
This contradicts the first equation, \(x + y - z = 7\), so the system is inconsistent and has no solution.