Question

If $\left|\begin{array}{ccc}1+x& 1-x& 1-x\\ 1-x& 1+x& 1-x\\ 1-x& 1-x& 1+x\end{array}\right|$ $=0$ then $x=$

• A. $1,1$
• B. $1,-1$
• C. $0,-1$
• D. $0,3$

Answer 1

The correct option is D $0,3$

$\left|\begin{array}{ccc}1+x& 1-x& 1-x\\ 1-x& 1+x& 1-x\\ 1-x& 1-x& 1+x\end{array}\right|=0$
$C_1\to C_1+C_2+C_3$
$\Rightarrow \left|\begin{array}{ccc}3-x& 1-x& 1-x\\ 3-x& 1+x& 1-x\\ 3-x& 1-x& 1+x\end{array}\right|=0$
$\Rightarrow (3-x)\left|\begin{array}{ccc}1& 1-x& 1-x\\ 1& 1+x& 1-x\\ 1& 1-x& 1+x\end{array}\right|=0$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$\Rightarrow (3-x)\left|\begin{array}{ccc}0& -2x& 0\\ 0& 2x& -2x\\ 1& 1-x& 1+x\end{array}\right|=0$
$\Rightarrow 4x^2(3-x)=0$
$\Rightarrow x=3,0$