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Question
Prove the following identity:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\]
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Solution
\[LHS\]
\[ = \begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
\[ = \begin{vmatrix}a + x + y + z & y & z \\ a + x + y + z & a + y & z \\ a + x + y + z & y & a + z\end{vmatrix} \left[\text{ Applying }C_1 \text{ to }C_1 + C_2 + C_3 \right]\]
\[ = \left( a + x + y + z \right)\begin{vmatrix}1 & y & z \\ 1 & a + y & z \\ 1 & y & a + z\end{vmatrix} \left[\text{ Taking }\left( a + x + y + z \right)\text{ common from }C_1 \right]\]
\[ = \left( a + x + y + z \right)\begin{vmatrix}1 & y & z \\ 0 & a & 0 \\ 0 & 0 & a\end{vmatrix} \left[\text{ Applying }R_2 \text{ to } R_2 - R_1\text{ and }R_3 \text{ to }R_3 - R_1 \right]\]
\[ = \left( a + x + y + z \right) a^2 \left[\text{ Expanding along first column }\right]\]
\[ = a^2 \left( a + x + y + z \right)\]
\[ = RHS\]
\[ \therefore \begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\]
