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Question
Solve the following determinant equation:
Solution
\[\text{ Let }∆ = \begin{vmatrix}x + a & x & x \\ x & x + a & x \\ x & x & x + a\end{vmatrix}\]
\[ = \begin{vmatrix}3x + a & x & x \\ 3x + a & x + a & x \\ 3x + a & x & x + a\end{vmatrix} \left[\text{ Applying }C_1 \text{ to }C_1 + C_2 + C_3 \right]\]
\[ = \left( 3x + a \right)\begin{vmatrix}1 & x & x \\ 1 & x + a & x \\ 1 & x & x + a\end{vmatrix}\]
\[ = \left( 3x + a \right)\begin{vmatrix}1 & x & x \\ 0 & a & 0 \\ 1 & x & x + a\end{vmatrix} \left[\text{ Applying }R_2 \text{ to }R_2 - R_1 \right]\]
\[ = \left( 3x + a \right)\begin{vmatrix}1 & x & x \\ 0 & a & 0 \\ 0 & 0 & a\end{vmatrix} \left[\text{ Applying }R_3 \text{ to }R_3 - R_1 \right]\]
\[ ∆ = \left( 3x + a \right)\left( a^2 - 0 \right) = 0\]
\[x = \frac{- a}{3}\]
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