Question

Evaluate: $\left|\begin{array}{ccc}1& bc& a(b+c)\\ 1& ca& b(c+a)\\ 1& ab& c(a+b)\end{array}\right|$

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

Answer 1

The correct option is A $0$

Performing $C_2\to \frac{C_2}{abc}{C}_3\to \frac{C_3}{abc}$ and then $C_3\to C_3+C_2$
$\left|\begin{array}{ccc}1& \frac{1}{a}& \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\\ 1& \frac{1}{b}& \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\\ 1& \frac{1}{c}& \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\end{array}\right|=(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\left|\begin{array}{ccc}1& \frac{1}{a}& 1\\ 1& \frac{1}{b}& 1\\ 1& \frac{1}{c}& 1\end{array}\right|=0$
(Two columns are identical on taking $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ common)