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Question
Show that \[\begin{vmatrix}y + z & x & y \\ z + x & z & x \\ x + y & y & z\end{vmatrix} = \left( x + y + z \right) \left( x - z \right)^2\]
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Solution
\[Let ∆ =| y + z x y\]
\[ z + x z x \]
\[x + y y z |\]
\[ \Rightarrow ∆ = | 2\left( x + y + z \right) x + y + z x + y + z\]
\[ z + x z x \]
\[ x + y y z t | \left[ \text{ Applying } R_1 \to R_1 + R_2 + R_3 \right]\]
\[ = \left( x + y + z \right) | 2 1 1 \]
\[ z + x z x \]
\[ x + y y z | \]
\[ = \left( x + y + z \right) 0 1 1\]
\[0 z x\]
\[ x - z y z | \left[ \text{ Applying } C_1 \to C_1 - C_2 - C_3 \right]\]
\[ = \left( x + y + z \right)\left\{ \left( x - z \right) \times \begin{vmatrix}1 & 1 \\ z & x\end{vmatrix} \right\} \left[ \text{ Expanding along } C_1 \right]\]
\[ = \left( x + y + z \right) \left( x - z \right)^2 \]
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