Question

The value of $\left|\begin{array}{ccc}a-b& b+c& a\\ b-a& c+a& b\\ c-a& a+b& c\end{array}\right|$

(a) $a^3+b^3+c^3$
(b) $3bc$
(c) $a^3+b^3+c^3-3abc$
(d) None of these

Answer 1

(d) We have,
$\left|\begin{array}{ccc}a-b& b+c& a\\ b-a& c+a& b\\ c-a& a+b& c\end{array}\right|=\left|\begin{array}{ccc}a+c& b+c+a& a\\ b+c& c+a+b& b\\ c+b& a+b+c& c\end{array}\right|$ $[\because C_1\to C_1+C_2\text{and}{C}_2\to C_2+C_3]$
$=(a+b+c)\left|\begin{array}{ccc}a+c& 1& a\\ b+c& 1& b\\ c+b& 1& c\end{array}\right|$ [taking (a+b+c) common from $C_2$]
$=(a+b+c)\left|\begin{array}{ccc}a-b& 0& a-c\\ 0& 0& b-c\\ c+b& 1& c\end{array}\right|$ $[\because R_2\to R_2-R_3\text{and}{R}_1\to R_1-R_3]$
$=(a+b+c)[-(b-c).(a-b\left)\right]$ [expanding along $R_2$]
$=(a+b+c)(c-b)(a-b)$