Question

Solve the equation $\left|\begin{array}{ccc}x+a& x& x\\ x& x+a& x\\ x& x& x+a\end{array}\right|=0,a\ne 0$

Answer 1

$\left|\begin{array}{ccc}x+a& x& x\\ x& x+a& x\\ x& x& x+a\end{array}\right|=0$
Applying $R_1=R_1-R_2,R_2=R_2-R_3$
$\left|\begin{array}{ccc}a& -a& 0\\ 0& a& -a\\ x& x& x+a\end{array}\right|=0$
Taking $a$ common from first and second row respectively
$\Rightarrow a^2\left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ x& x& x+a\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ x& x& x+a\end{array}\right|=0,\because a\ne 0$
Now expanding along first row we get,
$1(x+a+x)+1(0+x)=0$
$\Rightarrow 3x+a=0\Rightarrow x=-\frac{a}{3}$