| , * 20 c a - b - c & 2a & 2a 2b & b - c - a & 2b 2c & 2c & c - … - Vidyadip

MCQ

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

A
${(a + b + c)^2}$
✓
${(a + b + c)^3}$
C
$(a + b + c)(ab + bc + ca)$
D
None of these

✓

Answer

Correct option: B.

${(a + b + c)^3}$

$\left| {\,\begin{array}{*{20}{c}}{a - b - c}&{2a}&{2a}\\{2b}&{b - c - a}&{2b}\\{2c}&{2c}&{c - a - b}\end{array}\,} \right|$
$= \left| {\,\begin{array}{*{20}{c}}{ - \Sigma a}&0&{2a}\\{\Sigma a}&{ - \Sigma a}&{2b}\\0&{\Sigma a}&{c - a - b}\end{array}\,} \right| ,$
$\left( \begin{array}{l}{C_1} \to {C_1} - {C_2}\\{C_2} \to {C_2} - {C_3}\end{array} \right)$
$= {(\Sigma a)^2}\,\left| {\,\begin{array}{*{20}{c}}{ - 1}&0&{2a}\\1&{ - 1}&{2b}\\1&1&{c - a - b}\end{array}\,} \right| = {(\Sigma a)^3}, $
$($ on expansion$)$
$= {(a + b + c)^3}$.

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