Question

For how many values of x in the closed interval $[-4,-1]$ is the matrix $\left[\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x& -1& 2\end{array}\right]$ singular ?

• A. $2$
• B. $0$
• C. $3$
• D. $1$

Answer 1

The correct option is D $1$

Let , $A=\left[\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x& -1& 2\end{array}\right]$
For singular matrix,
$|A|=0$
$\Rightarrow \left|\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x& -1& 2\end{array}\right|=0$
$\Rightarrow x^2(x^2+x-3)=0$

$\Rightarrow x=0,x^2+x-3=0$
$\Rightarrow x=0,\frac{-1\pm \sqrt{13}}{2}$
But $0,\frac{-1+\sqrt{13}}{2}\notin [-4,-1]$
Hence, $x=\frac{-1-\sqrt{13}}{2}$
Hence, there is only one value of $x\in [-4,-1]$ for which given matrix is singular.