Question

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

Answer 1

Given matrix is singular. Hence,
$\left|\begin{array}{ccc}3& x-1& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|=0$
or
$\left|\begin{array}{ccc}0& x& -x\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|=0[R_1\to R_1-R_2]$
or
$\left|\begin{array}{ccc}0& x& -x\\ -x& 0& x\\ x+3& -1& 2\end{array}\right|=0[R_2\to R_2-R_3]$
or
$\left|\begin{array}{ccc}0& x& 0\\ -x& 0& x\\ x+3& -1& 1\end{array}\right|=0[C_3\to C_2-C_2]$
or $-x\left[\right(-x)-x(x+3\left)\right]=0$ or $x(x^2+4x)=0$ or $x=0$, -4
Hence, only one value of x is in closed interval $[-4,-1]$, i.e., $x=-4$.