Question

Prove that :

\[\begin{vmatrix}x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4\end{vmatrix} = 16 \left( 3x + 4 \right)\]

Answer 1

\[\text{ Let LHS }= \Delta = \begin{vmatrix} x + 4 & x & x\\x & x + 4 & x\\x & x & x + 4 \end{vmatrix}\] 
\[ = \begin{vmatrix} 3x + 4 & 3x + 4 & 3x + 4\\x & x + 4 & x\\x & x & x + 4 \end{vmatrix} \left[\text{ Applying }R_1 \to R_1 + R_2 + R_3 \right] \] 
\[ = \left( 3x + 4 \right)\begin{vmatrix} 1 & 1 & 1\\x & x + 4 & x \\x & x & x + 4 \end{vmatrix} \left[\text{ Taking out }\left( 3x + 4 \right)\text{ common from }R_1 \right]\] 
\[ = \left( 3x + 4 \right)\begin{vmatrix} 1 & 0 & 0\\x & 4 & 0\\x & 0 & 4 \end{vmatrix} \left[\text{ Applying }C_2 \to C_2 - C_1\text{ and }C_3 \to C_3 - C_1 \right]\] 
\[ = \left( 3x + 4 \right) \left( 4^2 \right) \left[\text{ Expanding along }R_1 \right]\] 
\[ = 16\left( 3x + 4 \right) \] 
\[ = RHS\]