Question

Determinant $\left|\begin{array}{ccc}1& a& a^2-bc\\ 1& b& b^2-ac\\ 1& c& c^2-ab\end{array}\right|$ is equal to

• A. $a+b+c$
• B. $a^2+b^2+c^2$
• C. 0
• D. 1

Answer 1

Answer: (C)

$\left|\begin{array}{ccc}1& a& a^2-bc\\ 1& b& b^2-ac\\ 1& c& c^2-ab\end{array}\right|$
$=\left|\begin{array}{ccc}1& a& a^2-bc\\ 0& b-a& b^2-ac-a^2+bc\\ 0& c-a& c^2-ab-a^2+bc\end{array}\right|$
$[R_2\to R_2-R_1,R_3\to R_3-R_1]$
$=1[(b-a)(c^2-ab-a^2+bc)-(c-a)$ $(b^2-ac-a^2+bc)]$
$=b{c}^2-a{b}^2-a^{2}b+b^{2}c-a{c}^2+a^{2}b$
$+a^3-abc-b^{2}c+a{c}^2+a^{2}c-b{c}^2+a{b}^2$
$-a^{2}c-a^3+abc$ $=0$