By using properties of determinants, prove the following: x + 4 &… - Vidyadip

Question

By using properties of determinants, prove the following:$\begin{vmatrix} \text {x + 4} & \text{2x} & \text{2x} \\ \text{ 2x} & \text{x + 4} & \text{2x} \\ \text{2x} & \text{2x} & \text{x + 4} \end{vmatrix} = ( 5\text{x} + 4) (4 -\text{x}) = 1. $

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Answer

$C_{1} \rightarrow C_{1} + C_{2} + C_{3} \text{gives Det} = (5\text{x} + 4) \begin{vmatrix} 1 & 2\text{x} & 2\text{x} \\ 1 & \text{x} + 4 & 2\text{x} \\ 1 & 2\text{x} & \text{x} + 4 \end{vmatrix} $$\begin{matrix} R_{2} \rightarrow & R_{2} - & R_{1} \\ R_{3} \rightarrow & R_{3} & R_{1} \\ \end{matrix} \Rightarrow \text{Det} = (5\text{x} + 4) \begin{vmatrix} 1 & 2\text{x} & 2\text{x} \\ 0 & 4 -\text{x} & 0 \\ 0 & 0 & 4 - \text{x} \end{vmatrix} $
$\text{ Expanding by C}_{1} \text{to get Det.} = ( 5\text{x} + 4) (4- \text{x})^{2}$

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