| , * 20 c 1&a& a^2 - bc 1&b& b^2 - ac 1&c& c^2 - ab , | = - Vidyadip

Question

$\left| {\,\begin{array}{*{20}{c}}1&a&{{a^2} - bc}\\1&b&{{b^2} - ac}\\1&c&{{c^2} - ab}\end{array}\,} \right| = $

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Answer

$\left| \,\begin{matrix} 1 & a & {{a}^{2}}-bc \\ 1 & b & {{b}^{2}}-ac \\ 1 & c & {{c}^{2}}-ab \\ \end{matrix}\,\right|$
$=\left| \,\begin{matrix} 0 & a-b & (a-b)\,(a+b+c) \\ 0 & b-c & (b-c)\,\,(a+b+c) \\ 1 & c &{{c}^{2}}-ab \\ \end{matrix}\, \right|$
by $\left\{ \begin{array}{l}{R_1} \to {R_1} - {R_2}\\{R_2} \to {R_2} - {R_3}\end{array} \right.$
$= (a - b)\,(b - c)\,\left| {\,\begin{array}{*{20}{c}}0&1&{a + b + c}\\0&1&{a + b + c}\\1&c&{{c^2} - ab}\end{array}\,} \right| = 0,$
$\{\because\,\,{R_1} \equiv {R_2}\} $

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