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.
Answer & explanation
Correct answer: option 4
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.