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