Question

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

Answer 1

Given, $\left|\begin{array}{ccc}x+a& x& x\\ x& x+a& x\\ x& x& x+a\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}3x+a& 3x+a& 3x+a\\ x& x+a& x\\ x& x& x+a\end{array}\right|=0$
(using $R_1\to R_1+R_2+R_3$)
$\Rightarrow (3x+a)\left|\begin{array}{ccc}1& 1& 1\\ x& x+a& x\\ x& x& x+a\end{array}\right|=0$
(Taking out $(3x+a)$ from $R_1$)
$\Rightarrow (3x+a)\left|\begin{array}{ccc}1& 0& 0\\ x& a& 0\\ x& a& 0\end{array}\right|=0$, (using $C_2\to C_2-C_1$ and $C_3\to C_3-C_1$)
Expanding corresponding to $R_1$ (first row), we get
$(3x+a)\left[1\right(a\times a-0\left)\right]=0\Rightarrow a^2(3x+a)=0$
But $a\ne 0$ (given). Therefore, $3x+a=0\Rightarrow x=-a/3$