If the system of equations
\(x-y+z=4\)
\(x-2y-2z=9\)
\(2x + y + \lambda z = 1\)
has a unique solution, then
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\lambda \ne 11\)
A system of three linear equations in three unknowns has a unique solution exactly when the determinant of its coefficient matrix is non-zero.
\(\displaystyle \Delta = \begin{vmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & \lambda \end{vmatrix}\)
Expanding along the first row:
\(\Delta = 1(-2\lambda + 2) + 1(\lambda + 4) + 1(1 + 4)\)
\(= -2\lambda + 2 + \lambda + 4 + 5\)
\(= 11 - \lambda\)
So \(\Delta \ne 0\) exactly when \(\lambda \ne 11\).