Prove the following using properties of determinants: a + b + 2c … - Vidyadip

Question

Prove the following using properties of determinants:
$ \begin{vmatrix} \text{a + b + 2c} & \text{a} & \text{b} \\ \text{c} & \text{b + c + 2a} & \text{b} \\ \text{c} & \text{a} & \text{c + a + 2b} \end{vmatrix}= 2(\text{a + b + c})^3 $

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Answer

$ \text{LHS}=\begin{vmatrix} 2(\text{a + b + c}) & \text{a} & \text{b} \\ 2(\text{a + b + c}) & \text{b + c + 2a} & \text{b} \\ 2(\text{a + b + c}) & \text{a} & \text{c + a + 2b} \end{vmatrix}\ \begin{matrix} \text{Using}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \text{C}_1\rightarrow\text{C}_1+\text{C}_2+\text{C}_3 \\ \end{matrix} $ $ \text{LHS}=\begin{vmatrix} 2(\text{a + b + c}) & \text{a} & \text{b} \\ 0 & \text{a + b + c} & 0 \\ 0 & 0 & \text{a + b + c} \end{vmatrix}\ \begin{matrix} \text{Using}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \text{R}_2\rightarrow\text{R}_2-\text{R}_1;\ \text{R}_3\rightarrow\text{R}_3-\text{R}_1 \\ \end{matrix}$
$= 2 (a + b + c) \{(a + b + c)^2 – 0\}$ Expanding along $C_1$
$= 2 (a + b + c)^3 = RHS$

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