Question

Number of real roots of the equation $\left|\begin{array}{ccc}1& x& x\\ x& 1& x\\ x& x& 1\end{array}\right|+\left|\begin{array}{ccc}1-x& 1& 1\\ 1& 1-x& 1\\ 1& 1& 1-x\end{array}\right|=0$ is

Answer 1

Answer: 1

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