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 find the value of x.

Answer 1

Given, $\left|\begin{array}{ccc}4-x& 4+x& 4+x\\ 4+x& 4-x& 4+x\\ 4+x& 4+x& 4-x\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}12+x& 12+x& 12+x\\ 4+x& 4-x& 4+x\\ 4+x& 4+x& 4-x\end{array}\right|=0$ $[\because R_1\to R_1+R_2+R_3]$
$\Rightarrow (12+x)\left|\begin{array}{ccc}1& 1& 1\\ 4+x& 4-x& 4+x\\ 4+x& 4+x& 4-x\end{array}\right|=0$ [taking (12+x) common from $R_1$]
$\Rightarrow (12+x)\left|\begin{array}{ccc}0& 0& 1\\ 0& -2x& 4+x\\ 2x& 2x& 4-x\end{array}\right|=0$ $[\because C_1\to C_1-C_3\text{and}{C}_2\to C_2-C_3]$
$\Rightarrow (12+x)\left[1\right(-(-4x^2)\left)\right]=0\Rightarrow (12+x)\left(4x^2\right)=0\therefore x=-12,0$