Question

Solve the equation $\begin{vmatrix} x+a &x &x \\\ x &x+a &x \\\ x&x &x+a \end{vmatrix}=0$ , a≠0

Answer 1

$\begin{vmatrix} x+a &x &x \\\ x &x+a &x \\\ x&x &x+a \end{vmatrix}=0$
Applying R1$\rightarrow$R1+R2+R3, we get:
$\begin{vmatrix} 3x+a &3x+a &3x+a \\\ x &x+a &x \\\ x&x &x+a \end{vmatrix}=0$

$\Rightarrow$(3x+a)$\begin{vmatrix} 1 &1 &1 \\\ x &x+a &x \\\ x&x &x+a \end{vmatrix}=0$

Applying C2$\rightarrow$C2-C1 and C3$\rightarrow$C3-C1, we have:
(3x+a)$\begin{vmatrix} 1&0 &0 \\\ x& a &0 \\\ x&0 &a \end{vmatrix}$=0

Expanding along R1,we have:
(3x+a)[1$\times$a2]=0
$\Rightarrow$a2(3x+a)=0
But a≠0,
Therefore we have
3x+a=0
$\Rightarrow x=-\frac{a}{3}$