Question

What is $\left|\begin{array}{ccc}-a^2& ab& ac\\ ab& -b^2& bc\\ ac& bc& -c^2\end{array}\right|$ equal to?

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

Answer 1

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

Taking out $a$ common in first row,$b$ common in second row and $c$ common in the third row we have,

$abc\left[\begin{array}{ccc}-a& b& c\\ a& -b& c\\ a& b& -c\end{array}\right]$

Now $R_2\to R_2+R_1$

$R_3\to R_3+R_1$

$abc\left[\begin{array}{ccc}-a& b& c\\ 0& 0& 2c\\ 0& 2b& 0\end{array}\right]$ opening the determinant

$abc\times a\left(4bc\right)$

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