For what value of x, the matrix $\left[\begin{array}{ccc}3-x & 2 & 2 \\ 2 & 4-x & 1 \\ -2 & -4 & -1-x\end{array}\right]$ is singular.

x = 0, 3

Since, the given matrix is singular

$\Rightarrow\left|\begin{array}{ccc}3-x & 2 & 2 \\ 2 & 4-x & 1 \\ -2 & -4 & -1-x\end{array}\right|=0$

$R_2+R_3$

$\Rightarrow\left|\begin{array}{ccc}3-x & 2 & 2 \\ 0 & -x & -x \\ -2 & -4 & -1-x\end{array}\right|=0$

$\Rightarrow x\left|\begin{array}{ccc}3-x & 2 & 2 \\ 0 & 1 & 1 \\ 2 & 4 & 1+x\end{array}\right|=0$

⇒ x {(3 – x) (1 + x – 4) – 0 + 2 (2 – 2)} = 0

⇒ x (3 – x) (x – 3) = 0 ⇒ x = 0, 3

Hence (4) is the correct answer.