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Question
Solve the following determinant equation:
Solution
\[\text{ Let }∆ = \begin{vmatrix}x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4\end{vmatrix}\]
\[ = \begin{vmatrix}x + 9 & 3 & 5 \\ x + 9 & x + 2 & 5 \\ x + 9 & 3 & x + 4\end{vmatrix} \left[\text{ Applying }C_1 = C_1 + C_2 + C_3 \right]\]
\[ = \left( x + 9 \right)\begin{vmatrix}1 & 3 & 5 \\ 1 & x + 2 & 5 \\ 1 & 3 & x + 4\end{vmatrix} \]
\[ = \left( x + 9 \right)\begin{vmatrix}1 & 3 & 5 \\ 0 & x - 1 & 0 \\ 1 & 3 & x + 4\end{vmatrix} \left[\text{ Applying }R_2 \text{ to } R_2 - R_1 \right]\]
\[ = \left( x + 9 \right)\begin{vmatrix}1 & 3 & 5 \\ 0 & x - 1 & 0 \\ 0 & 0 & x - 1\end{vmatrix} \left[\text{ Applying }R_3 \text{ to } R_3 - R_1 \right]\]
\[ ∆ = \left( x + 9 \right) \left( x - 1 \right)^2 = 0\]
\[x = - 9, 1, 1\]
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