Question

If a, b, & c are nonzero real numbers, then $\left|\begin{array}{ccc}{b}^2{c}^2& bc& b+c\\ c^2{a}^2& ca& c+a\\ a^2{b}^2& ab& a+b\end{array}\right|$ is equal to

• A. $a^2{b}^2{c}^2(a+b+c)$
• B. $abc(a+b+c{)}^2$
• C. zero
• D. none of these

Answer 1

The correct option is C zero

Multiplying $R_1,R_2,R_3$ by $a,b,c$ respectivley we get

$D=\frac{1}{abc}\left|\begin{array}{ccc}a{b}^2{c}^2& abc& a(b+c)\\ b{c}^2{a}^2& bca& b(c+a)\\ c{a}^2{b}^2& cab& c(a+b)\end{array}\right|$
$(R_1\to a{R}_1,R_2\to b{R}_2,R_3\to c{R}_3)$
$=abc\left|\begin{array}{ccc}bc& 1& ab+ac\\ ca& 1& bc+ab\\ ab& 1& ca+cb\end{array}\right|$
$=abc(ab+bc+ca)\left|\begin{array}{ccc}bc& 1& 1\\ ca& 1& 1\\ ab& 1& 1\end{array}\right|(C_3\to C_3+C_1)=0$