Question

The value of the determinant $\left|\begin{array}{ccc}-a& b& c\\ a& -b& c\\ a& b& -c\end{array}\right|$ is equal to-

• A. $0$
• B. $(a-b)(b-c)(c-a)$
• C. $(a+b)(b+c)(c+a)$
• D. $4abc$

Answer 1

Answer: (D)

$4abc$

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

Take $a,b,c$ as a common factor from $C_1,C_2,C_3$ respectively

$\Delta =\left(abc\right)\left|\begin{array}{ccc}-1& 1& 1\\ 1& -1& 1\\ 1& 1& -1\end{array}\right|$

Apply $C_2\to C_2+C_1$ and $C_3\to C_3+C_1$

$\Delta =\left(abc\right)\left|\begin{array}{ccc}-1& 0& 0\\ 1& 0& 2\\ 1& 2& 0\end{array}\right|$

Expanding the above matrix along $R_1$

$\Delta =\left(abc\right)[-1(0-(2\times 2))]$

$\Delta =4abc$

Hence, the value of the given determinant is $4abc$.