Question

If $\left|\begin{array}{ccc}4-x& 4+x& 4+x\\ 4+x& 4-x& 4+x\\ 4+x& 4+x& 4-x\end{array}\right|=0$,
then the sum of value(s) of $x$ is

• A. $0$
• B. $12$
• C. $-12$
• D. $24$

Answer 1

The correct option is C $-12$

We have $\left|\begin{array}{ccc}4-x& 4+x& 4+x\\ 4+x& 4-x& 4+x\\ 4+x& 4+x& 4-x\end{array}\right|=0$
Applying $R_1\to R_1+R_2+R_3$ $\Rightarrow \left|\begin{array}{ccc}12+x& 12+x& 12+x\\ 4+x& 4-x& 4+x\\ 4+x& 4+x& 4-x\end{array}\right|=0$
Taking $(12+x)$ common from $R_1$
$\Rightarrow (12+x)\left|\begin{array}{ccc}1& 1& 1\\ 4+x& 4-x& 4+x\\ 4+x& 4+x& 4-x\end{array}\right|=0$
Applying $C_1\to C_1-C_3$ and $C_2\to C_2-C_3$
$\Rightarrow (12+x)\left|\begin{array}{ccc}0& 0& 1\\ 0& -2x& 4+x\\ 2x& 2x& 4-x\end{array}\right|=0\Rightarrow (12+x)(0-(-2x\left)\right(2x)=0\Rightarrow (12+x)\left(4x^2\right)=0\therefore x=-12,0$