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