Question

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

• A. always positive
• B. always negative
• C. always zero
• D. cannot say anything

Answer 1

Answer: (A)

always positive

Given determinant is,

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

Consider the matrix $A=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

Cofactor of this matrix A is $A_C=\left|\begin{array}{ccc}bc-a^2& ac-b^2& ab-c^2\\ ac-b^2& ab-c^2& bc-a^2\\ ab-c^2& bc-a^2& ac-b^2\end{array}\right|$

Thus, we can say that cofactor of matrix A is given matrix D.

Thus, by property of cofactor of matrix, we can write,

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

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

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

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

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

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