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Evaluate the Following: ∣ ∣ ∣ ∣ a + X Y Z X a + Y Z X Y a + Z ∣ ∣ ∣ ∣

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Question

Evaluate the following:

\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]

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Solution

Let 

\[∆ = \begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]

\[∆ = \begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]

\[ = \begin{vmatrix}a + x + y + z & y & z \\ a + x + y + z & a + y & z \\ a + x + y + z & y & a + z\end{vmatrix} \left[\text{ Applying }C_1 \text{ to }C_1 + C_2 + C_3 \right]\]

\[ = \left( a + x + y + z \right)\begin{vmatrix}1 & y & z \\ 1 & a + y & z \\ 1 & y & a + z\end{vmatrix} \left[\text{ Taking }\left( a + x + y + z \right)\text{ common from }C_1 \right]\]

\[ = \left( a + x + y + z \right)\begin{vmatrix}1 & y & z \\ 0 & a & 0 \\ 0 & 0 & a\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \text{ and }R_3 \to R_3 - R_1 \right]\]

\[ = \left( a + x + y + z \right) a^2 \left[\text{ Expanding along first column }\right]\]

\[ \therefore ∆ = \left( a + x + y + z \right) a^2\]

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Chapter 5: Determinants - Exercise 6.2 [Page 58]

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R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12
Chapter 5 Determinants
Exercise 6.2 | Q 9 | Page 58

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