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

• A. $2$
• B. $3$
• C. $0$
• D. $1$

Answer 1

Answer: (D)

$1$

Since inverse not exist

$\left[\begin{array}{ccc}1& -1& x\\ 1& x& 1\\ x& -1& 1\end{array}\right]=0$

$\left[\begin{array}{ccc}1& -1& x\\ 0& x+1& 1-x\\ x& -1& 1\end{array}\right]=0$

Expanding along $C_1$

$1[x+1+1-x]+x[-1+x-x^2-x]=0$

$x^3+x-2=0$

$(x-1)(x^2+x+2)=0$

$x=1$ ($x^2+x+2$ has imaginary roots)