Question

Number of value(s) of $x$ in the interval $[-4,-1]$ for which the matrix $A$ is singular.
$A=\left[\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right]$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B $1$

For a singular matrix, the value of determinant should be zero

$|A|=\left|\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|=0$
Applying, $R_2\to R_2-R_1,R_3\to R_3-R_1$
$|A|=\left|\begin{array}{ccc}3& -1+x& 2\\ 0& -x& x\\ x& -x& 0\end{array}\right|=0$
$\Rightarrow 3(0+x^2)+0+x(-x+x^2+2x)=0$
$\Rightarrow x=0,-4$
$\Rightarrow x=-4$ ($\because x=0$ does not lie in the given interval)
Therefore, only one value lies in the given interval.

Hence, option B.