Question

If a,b and care non zero numbers, then $\Delta =\left|\begin{array}{ccc}{b}^2{c}^2& bc& b+c\\ c^2{a}^2& ca& c+a\\ a^2{b}^2& ab& a+b\end{array}\right|$ is equal to

• A. abc
• B. $a^2{b}^2{c}^2$
• C. ab+bc+ca
• D. None of these

Answer 1

The correct option is D None of these

Multiplying $R_1$ by $a,R_2$ by b and $R_3$ by c, we have
$\Delta =\frac{1}{abc}\left|\begin{array}{ccc}a{b}^2{c}^2& abc& ab+ac\\ a^{2}b{c}^2& abc& bc+ab\\ a^2{b}^{2}c& abc& ac+bc\end{array}\right|=\frac{a^2{b}^2{c}^2}{abc}\left|\begin{array}{ccc}bc& 1& ab+ac\\ ac& 1& bc+ab\\ ab& 1& ac+bc\end{array}\right|=abc\left|\begin{array}{ccc}bc& 1& \sum ab\\ ac& 1& \sum ab\\ ab& 1& \sum ab\end{array}\right|\{by{C}_3\to C_3+C_1\}=abc.\sum ab\left|\begin{array}{ccc}bc& 1& 1\\ ca& 1& 1\\ ab& 1& 1\end{array}\right|=0,[Since{C}_2\equiv C_3]$.
Trick : Put a=1, b=2, c=3 and check it.