Question

If $\left[\begin{array}{ccc}1& -1& x\\ 1& x& 1\\ x& -1& 1\end{array}\right]$ has no inverse, then the possible real value of $x$ is

• A. $2$
• B. $3$
• C. $0$
• D. $1$

Answer 1

The correct option is D $1$

Let $A=\left[\begin{array}{ccc}1& -1& x\\ 1& x& 1\\ x& -1& 1\end{array}\right]$ has no inverse.
$\Rightarrow |A|=0,$ i.e., $\left|\begin{array}{ccc}1& -1& x\\ 1& x& 1\\ x& -1& 1\end{array}\right|=0$
$\Rightarrow 2-x-x^3=0$
$\Rightarrow (x-1)(x^2+x+2)=0$
$\Rightarrow x=1$