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

MCQ

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

✓
$0$
B
${a^3} + {b^3} + {c^3} - 3abc$
C
$3abc$
D
${(a + b + c)^3}$

✓

Answer

Correct option: A.

$0$

a
(a) $\left| \,\begin{matrix}
1 & a & {{a}^{2}}-bc \\
1 & b & {{b}^{2}}-ac \\
1 & c & {{c}^{2}}-ab \\
\end{matrix}\, \right|=\left| \,\begin{matrix}
0 & a-b & (a-b)\,(a+b+c) \\
0 & b-c & (b-c)\,\,(a+b+c) \\
1 & c & {{c}^{2}}-ab \\
\end{matrix}\, \right|$

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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