Question

If $a,b$ and $c$ are non-zero numbers, then the value of the determinant, $D=\left|\begin{array}{ccc}{b}^2{c}^2& bc& b+c\\ c^2{a}^2& ca& c+a\\ a^2{b}^2& ab& a+b\end{array}\right|$ is

• A. $abc$
• B. $a^2{b}^2{c}^2$
• C. $bc+ca+ab$
• D. 0

Answer 1

The correct option is D

0

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

$=\frac{1}{abc}\left|\begin{array}{ccc}a{b}^2{c}^2& abc& a(b+c)\\ b{c}^2{a}^2& abc& b(a+c)\\ a^2{b}^2& abc& c(a+b)\end{array}\right|$

$=\frac{abc.abc}{abc}\left|\begin{array}{ccc}bc& 1& a(b+c)\\ ac& 1& b(a+c)\\ ab& 1& c(a+b)\end{array}\right|$

$C_3\to C_3+C_1$

$=abc\left|\begin{array}{ccc}bc& 1& ab+bc+ca\\ ac& 1& ab+bc+ca\\ ab& 1& ab+bc+ca\end{array}\right|$

$=abc(ab+bc+ca)\left|\begin{array}{ccc}bc& 1& 1\\ ac& 1& 1\\ ab& 1& 1\end{array}\right|=0$