Using properties of determinants, prove the following: a & b & c … - Vidyadip

Question

Using properties of determinants, prove the following:
$\begin{vmatrix}\text{a} & \text{b} & \text{c} \\\text{a}-\text{b} & \text{b}-\text{c} & \text{c}-\text{a}\\\text{b}+\text{c} & \text{c}+\text{a} & \text{a}+\text{b} \end{vmatrix}=\text{a}^3+\text{b}^3+\text{c}^3-3\text{abc}.$

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Answer

LHS: Let $\triangle=\begin{vmatrix}\text{a} & \text{b} & \text{c} \\\text{a}-\text{b} & \text{b}-\text{c} & \text{c}-\text{a}\\\text{b}+\text{c} & \text{c}+\text{a} & \text{a}+\text{b} \end{vmatrix}$
APPLY $C_1 → C_1 + C_2 + C_3$
$=\begin{vmatrix}\text{a}+\text{b}+\text{c} & \text{b} & \text{c} \\0 & \text{b}-\text{c} & \text{c}-\text{a}\\2(\text{a}+\text{b}+\text{c}) & \text{c}+\text{a} & \text{a}+\text{b} \end{vmatrix}$
Taking (a + b + c) common from $C_1$
$=(\text{a}+\text{b}+\text{c})\begin{vmatrix}1 & \text{b} & \text{c} \\0 & \text{b}-\text{c} & \text{c}-\text{a}\\2 & \text{c}+\text{a} & \text{a}+\text{b} \end{vmatrix}$
Apply: $R_3 → R_3 - 2R_1$
$=(\text{a}+\text{b}+\text{c})\begin{vmatrix}1 & \text{b} & \text{c} \\0 & \text{b}-\text{c} & \text{c}-\text{a}\\0 & \text{c}+\text{a} -\text{2b}& \text{a}+\text{b} -2\text{c}\end{vmatrix}$
$=(\text{a}+\text{b}+\text{c})[(\text{b}-\text{c})(\text{a}+\text{b}-2\text{c})-(\text{c}-\text{a})(\text{c}+\text{a}-2\text{b})]$
$=\text{a}^3+\text{b}^3+\text{c}^{3}-3\text{abc}$
$=\text{RHS}$

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