Evaluate […] .

The correct answer is Option (3) → $a^2(a + x + y + z)$ ##

We have, $\begin{vmatrix} a + x & y & z \\ x & a + y & z \\ x & y & a + z \end{vmatrix} = \begin{vmatrix} a & -a & 0 \\ 0 & a & -a \\ x & y & a + z \end{vmatrix} \quad \left[ \begin{aligned} &∵R_1 \to R_1 - R_2 \\ &\text{and } R_2 \to R_2 - R_3 \end{aligned} \right]$

$= \begin{vmatrix} a & 0 & 0 \\ 0 & a & -a \\ x & x + y & a + z \end{vmatrix} \quad [∵ C_2 \to C_2 + C_1]$

On expanding along $R_1$, we get

$= a(a^2 + az + ax + ay)$

$= a^2 (a + z + x + y)$