Question

The value of $\left|\begin{array}{ccc}1& bc& a(b+c)\\ 1& ac& b(a+c)\\ 1& ab& c(a+b)\end{array}\right|$ is, $($where $a\ne b\ne c)$

• A. $(a-b)(b-c)(c-a)$
• B. $0$
• C. $(a+b+c)(a^2+b^2+c^2)$
• D. $1$

Answer 1

The correct option is B $0$

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

Using $C_3\to C_3+C_2:$
we have,
$\Delta =\left|\begin{array}{ccc}1& bc& ab+bc+ac\\ 1& ac& ba+bc+ac\\ 1& ab& ca+bc+ab\end{array}\right|$
Taking $ab+bc+ac$ common from $C_3,$ we get
$=(ab+bc+ac)\left|\begin{array}{ccc}1& bc& 1\\ 1& ac& 1\\ 1& ab& 1\end{array}\right|=(ab+bc+ac)\times 0=0$
$(\because C_1=C_3)$