Using the properties of determinants, prove that a + b & b + c & … - Vidyadip

Question

Using the properties of determinants, prove that
$ \begin{vmatrix} \text{a + b} & \text{b + c} & \text{c + a} \\ \text{b + c} & \text{c + a} & \text{a + b} \\ \text{c + a} & \text{a + b} & \text{b + c} \end{vmatrix}=2 \begin{vmatrix} \text{a} & \text{b} & \text{c} \\ \text{b} & \text{c} & \text{a} \\ \text{c} & \text{a} & \text{b} \end{vmatrix}$.

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Answer

Using $C_1 \rightarrow C_1 + C_2 + C_3$ we get LHS = 2 $ \begin{vmatrix} \text{a + b + c} & \text{b + c} & \text{c + a} \\ \text{a + b + c} & \text{c + a} & \text{a + b} \\ \text{a + b + c} & \text{a + b} & \text{b + c} \end{vmatrix}$
$C_1 \rightarrow C_1 – C_2$ to get = 2 $ \begin{vmatrix} \text{a } & \text{b + c} & \text{c + a} \\ \text{b } & \text{c + a} & \text{a + b} \\ \text{c } & \text{a + b} & \text{b + c} \end{vmatrix}$

Using $C_3 \rightarrow C_3 – C_1 = 2 \begin{vmatrix} \text{a } & \text{b + c} & \text{c} \\ \text{b } & \text{c + a} & \text{a} \\ \text{c } & \text{a + b} & \text{b } \end{vmatrix}$
Using $C_2 \rightarrow C_2 – C_3 = 2 \begin{vmatrix} \text{a } & \text{b} & \text{c} \\ \text{b } & \text{c } & \text{a} \\ \text{c } & \text{a } & \text{b } \end{vmatrix}=\text{RHS}$.

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