Evaluate $\begin{vmatrix} x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4 \end{vmatrix}$.

$16(3x + 4)$

The correct answer is Option (1) → $16(3x + 4)$ ##

We have, $\begin{vmatrix} x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4 \end{vmatrix}$

On applying, $R_1 \to R_1 + R_2 + R_3$, we get

$= \begin{vmatrix} 3x + 4 & 3x + 4 & 3x + 4 \\ x & x + 4 & x \\ x & x & x + 4 \end{vmatrix}$

On taking $(3x + 4)$ common from First row, we get

$= (3x + 4) \begin{vmatrix} 1 & 1 & 1 \\ x & x + 4 & x \\ x & x & x + 4 \end{vmatrix}$

Now, on applying $C_2 \to C_2 - C_1, C_3 \to C_3 - C_1$, we get

$= (3x + 4) \begin{vmatrix} 1 & 0 & 0 \\ x & 4 & 0 \\ x & 0 & 4 \end{vmatrix}$

On expanding along First row, we get

$= (3x + 4) [1(4 \times 4 - 0 \times 0)]$

$= 16 \cdot (3x + 4)$