Question

What is the value of the determinant $\left|\begin{array}{ccc}1& bc& a(b+c)\\ 1& ca& b(c+a)\\ 1& ab& c(a+b)\end{array}\right|$?

• A. $0$
• B. $abc$
• C. $ab+bc+ca$
• D. $abc(a+b+c)$

Answer 1

The correct option is A $0$

Now, in the matrix

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

$C_3\to C_3+C_2$

$\left|\begin{array}{ccc}1& bc& ab+ac+bc\\ 1& ac& ab+ac+bc\\ 1& ab& ab+ac+bc\end{array}\right|$ taking $(ab+bc+ac)$ common in $C_3$

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

Now two columns of the matrix are same so it's determinant will be 0.

(a)