Question

If $a,b,$ and $c$ are nonzero real 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. $bc+ca+ab$
• D. none of these

Answer 1

Answer: (D)

Applying $R_1\to a{R}_1,R_2\to b{R}_2$ and $R_3\to c{R}_3$, we get
$\Delta =\frac{1}{abc}\left|\begin{array}{ccc}a{b}^2{c}^2& abc& ab+ac\\ a^{2}b{c}^2& abc& bc+ab\\ a^2{b}^{2}c& abc& ac+bc\end{array}\right|=\frac{a^2{b}^2{c}^2}{abc}\left|\begin{array}{ccc}bc& 1& ab+ac\\ ac& 1& bc+ab\\ ab& 1& ac+bc\end{array}\right|$
Applying $C_3\to C_3+C_1$ and taking $\left(bc+ca+ab\right)$ common, we get
$\Delta =abc\left(bc+ca+ab\right)\left|\begin{array}{ccc}bc& 1& 1\\ ac& 1& 1\\ ab& 1& 1\end{array}\right|=0$
$\left[\because C_2\text{and}{C}_3\text{are identical}\right]$