If the system of equation
$x-y+z=4$
$x-2y-2z=9$
$2x + y + λz = 1$
has a unique solution, then

$λ≠11$

The correct answer is Option (3) → $λ≠11$

Coefficient matrix: $\begin{bmatrix}1&-1&1\\1&-2&-2\\2&1&\lambda\end{bmatrix}$

$\Delta=\begin{vmatrix}1&-1&1\\1&-2&-2\\2&1&\lambda\end{vmatrix} =1\begin{vmatrix}-2&-2\\1&\lambda\end{vmatrix}+1\begin{vmatrix}1&-2\\2&\lambda\end{vmatrix}+1\begin{vmatrix}1&-2\\2&1\end{vmatrix}$

$=(-2\lambda+2)+(\lambda+4)+5=11-\lambda$

Unique solution $\iff \Delta\ne 0 \iff 11-\lambda\ne 0$

$\lambda \ne 11$