Question

The value of $\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|$, is

• A. $6abc$
• B. $a+b+c$
• C. $4abc$
• D. $abc$

Answer 1

The correct option is C $4abc$

$\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|$
$R_1\to R_1+R_2+R_3$
$\left|\begin{array}{ccc}2(b+c)& 2(a+c)& 2(a+b)\\ b& c+a& b\\ c& c& a+b\end{array}\right|$
$=2\left|\begin{array}{ccc}b+c& a+c& a+b\\ b& c+a& b\\ c& c& a+b\end{array}\right|$
$R_1\to R_1-R_3$
$=2\left|\begin{array}{ccc}b& a& 0\\ b& c+a& b\\ c& c& a+b\end{array}\right|$
$R_2\to R_2-R_1$
$=2\left|\begin{array}{ccc}b& a& 0\\ 0& c& b\\ c& c& a+b\end{array}\right|$
On expanding the above determinant we get,
$=2(abc+abc)=4abc$
Hence, option 'C' is correct.