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  2. Solve the equation determinant x + a x x x x + a x x x x + a equals 0; a not equal to 0.
# NCERT

Q : 5        Solve the equation     

                 \begin{vmatrix} x+a & x &x \\ x &x+a &x\\ x & x & x+a \end{vmatrix}=0; a\neq 0

Answers (1)
D Divya Prakash Singh

Given determinant \begin{vmatrix} x+a & x &x \\ x &x+a &x\\ x & x & x+a \end{vmatrix}=0; a\neq 0

Applying the row transformation; R_{1} \rightarrow R_{1}+R_{2}+R_{3} we have;

\begin{vmatrix} 3x+a & 3x+a &3x+a \\ x &x+a &x\\ x & x & x+a \end{vmatrix} =0

Taking common factor (3x+a) out from first row.

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

Now applying the column transformations; C_{1} \rightarrow C_{1}-C_{2} and C_{2} \rightarrow C_{2}-C_{3}.

we get;

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

=(3x+a)(a^2)=0           as    a^2 \neq 0,

or 3x+a=0  or   x= -\frac{a}{3}

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