Solve the equation \(\begin{vmatrix} x+a &x &x \\ x &x+a &x \\ x&x &x+a \end{vmatrix}=0\) , a≠0
Solve the equation \(\begin{vmatrix} x+a &x &x \\ x &x+a &x \\ x&x &x+a \end{vmatrix}=0\) , a≠0
Solution and Explanation
\(\begin{vmatrix} x+a &x &x \\ x &x+a &x \\ x&x &x+a \end{vmatrix}=0\)
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
\(\Rightarrow x=-\frac{a}{3}\)
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