Question

The value of $\left|\begin{array}{ccc}-a^2& ab& ac\\ ab& -b^2& bc\\ ac& bc& -c^2\end{array}\right|$ is

• A. $0$
• B. $abc$
• C. $4a^2{b}^2{c}^2$
• D. None of these

Answer 1

Answer: (C)

Let $\Delta =\left|\begin{array}{ccc}-a^2& ab& ac\\ ab& -b^2& bc\\ ac& bc& -c^2\end{array}\right|$
Taking $a,b,c$ common from $R_1,R_2$ and $R_3$ respectively, we get. $\Delta =abc\left|\begin{array}{ccc}-a& b& c\\ a& -b& c\\ a& b& -c\end{array}\right|]=a^2{b}^2{c}^2[\left|\begin{array}{ccc}-1& 1& 1\\ 1& -1& 1\\ 1& 1& -1\end{array}\right|$
taking a, b, c common from $C_1,C_2,C_3$ respectively]
$=a^2{b}^2{c}^2\left|\begin{array}{ccc}-1& 0& 0\\ 1& 0& 2\\ 1& 2& 0\end{array}\right|$
( applying $C_2\to C_2+C_1,C_3\to C_3+C_1)$
$=a^2{b}^2{c}^2\cdot (-1)\left|\begin{array}{cc}0& 2\\ 2& 0\end{array}\right||=a^2{b}^2{c}^2(-1)(0-4)$
$\Rightarrow \Delta =4a^2{b}^2{c}^2$