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Question
Evaluate the following:
\[\begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}\]
Solution
Let
\[∆ = \begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}\]
\[∆ = \begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}\]
\[ = \begin{vmatrix}x - 1 & 1 - x & 0 \\ 1 & x & 1 \\ 0 & 1 - x & x - 1\end{vmatrix} \left[\text{ Applying }R_1 \to R_1 - R_2\text{ and }R_3 \to R_3 - R_2 \right]\]
\[ = \left( x - 1 \right)^2 \begin{vmatrix}1 & - 1 & 0 \\ 1 & x & 1 \\ 0 & - 1 & 1\end{vmatrix}\]
\[ = \left( x - 1 \right)^2 \begin{vmatrix}1 & - 1 & 0 \\ 1 & x + 1 & 1 \\ 0 & 0 & 1\end{vmatrix} \left[\text{ Applying }C_2 \to C_2 + C_3 \right]\]
\[ = \left( x - 1 \right)^2 (x + 1 + 1) \left[\text{ Expanding along last row }\right]\]
\[ = \left( x - 1 \right)^2 (x + 2)\]
\[ \therefore ∆ = \left( x - 1 \right)^2 (x + 2)\]
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