Question

If $\left|\begin{array}{ccc}-a^2& ab& ac\\ ab& -b^2& bc\\ ac& bc& -c^2\end{array}\right|=k{a}^2{b}^2{c}^2$, then k is equal to

• A. $2$
• B. $4$
• C. $-4$
• D. $8$

Answer 1

The correct option is B $4$

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 $R_1,R_2,R_3$ respectively

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

And taking $a,b,c$ common from $C_1,C_2,C_3$

$\Delta =a^2{b}^2{c}^2\left|\begin{array}{ccc}-1& 1& 1\\ 1& -1& 1\\ 1& 1& -1\end{array}\right|$

$=a^2{b}^2{c}^2\{-1(1-1)-1(-1-1)+1(1+1)\}$

$=4a^2{b}^2{c}^2$