Question

The number of values of $x$ in $\left[-4,-1\right]$ , for which the matrix $\left[\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right]$ is singular, is

• A. $1$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Answer: (A)

For the matrix to be singular, its determinant value should be zero
$\left|\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|=0$
$\Rightarrow 3\left(-2+x+2\right)-\left(x-1\right)\left[6-\left(x+2\right)\left(x+3\right)\right]+2\left(-3+x+3\right)=0$
$\Rightarrow 3x+\left(x-1\right)x\left(x+5\right)+2x=0$
$\Rightarrow x\left[5+x^2+4x-5\right]=0\Rightarrow x=0,-4$
But, $x=0$ is rejected
Hence, only one value in $\left[-4,-1\right]$