Question

If $a,b,c\in R\text{and}\left|\begin{array}{ccc}x& -c& b\\ c& x& -a\\ -b& a& x\end{array}\right|=0$, then the number of real solutions of $x$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B $1$

$\left|\begin{array}{ccc}x& -c& b\\ c& x& -a\\ -b& a& x\end{array}\right|=0\Rightarrow x^3+x(a^2+b^2+c^2)=0\Rightarrow x=0\text{or}{x}^2=-(a^2+b^2+c^2)\Rightarrow \text{Only one real value of}x.$