Question

$\left|\begin{array}{ccc}a+b& a& b\\ a& a+c& c\\ b& c& b+c\end{array}\right|$ is equal to.

• A. $4abc$
• B. $abc$
• C. $a^2{b}^2{c}^2$
• D. $4a^{2}bc$

Answer 1

Answer: (A)

$4abc$

$\left|\begin{array}{ccc}a+b& a& b\\ a& a+c& c\\ b& c& b+c\end{array}\right|=\left|\begin{array}{ccc}2(a+b)& 2(a+c)& 2(b+c)\\ a& a+c& c\\ b& c& b+c\end{array}\right|,R_1\to R_1+R_2+R_3$

$=2\left|\begin{array}{ccc}a+b& a+c& b+c\\ a& a+c& c\\ b& c& b+c\end{array}\right|=2\left|\begin{array}{ccc}b& 0& b\\ a& a+c& c\\ b& c& b+c\end{array}\right|$$,R_1\to R_1-R_2$
$=2b\left|\begin{array}{ccc}1& 0& 1\\ a-b& a& -b\\ b& c& b+c\end{array}\right|,R_2\to R_2-R_3$

$=2b\left[a\right(b+c)+bc+ac-bc-ab]$, Expand along first row

$=2b\left(2ac\right)=4abc$