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
\(\Rightarrow x=-\frac{a}{3}\)