Question

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

Answer 1

If $A$ is singular then it's inverse doesn't exist$\Rightarrow$ det$\left(A\right)=0$
$\Rightarrow 1\cdot (x+1)+1\cdot (1+x)+x(-1+x^2)=0$
$\Rightarrow x^3+x+2=0$
$\Rightarrow (x+1)(x^2-x+2)=0$
$\therefore x=-1$
(as $x^2-x+2\ne 0$ for any value of $x.$)
$\Rightarrow |x|=1$