The value of determinant

Option (C) a3 + b3 + c3 – 3abc

Given,

$\Delta=\left|\begin{array}{lll}a-b & b+c & a \\ b-c & c+a & b \\ c-a & a+b & c\end{array}\right|$

[Applying $C_{1} \rightarrow C_{1}-C_{3}$ ]

$=\left|\begin{array}{lll}-b & b+c & a \\ -c & c+a & b \\ -a & a+b & c\end{array}\right|$

[Applying $C_{2} \rightarrow C_{2}+C_{1}$ ]

$=\left|\begin{array}{ccc}-b & c & a \\ -c & a & b \\ -a & b & c\end{array}\right|=-\left|\begin{array}{lll}b & c & a \\ c & a & b \\ a & b & c\end{array}\right|$

$=-\left[b\left(a c-b^{2}\right)-c\left(c^{2}-a b\right)+a\left(b c-a^{2}\right)\right]$

$=a^{3}+b^{3}+c^{3}-3 a b c$