Question

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

• A. $4abc$
• B. $4a^2{b}^2{c}^2$
• C. $a^2{b}^2{c}^2$
• D. $a^2+b^2+c^2-ab-bc-ca$

Answer 1

The correct option is B $4a^2{b}^2{c}^2$

Let $A=\left|\begin{array}{ccc}-a^2& ab& ac\\ ab& -b^2& bc\\ ac& bc& -c^2\end{array}\right|$

take common from rows

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

No, take common from columns,

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

By operation of matrix

$A=\left(abc{)}^2\right[-1(1-1)-1(-1-1)+1(1+1)]$

$A=\left(abc{)}^2\right(2+2)$

$\Rightarrow A=4a^2{b}^2{c}^2$