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