Question

If $\left|\begin{array}{ccc}a+b& b+c& c+a\\ c+a& a+b& b+c\\ b+c& c+a& a+b\end{array}\right|=t\times det$ of circulant matrix whose elements of first column are $a,b,c$ then $t$ equals

• A. 5
• B. 6
• C. -2
• D. 2

Answer 1

The correct option is C -2

$\left|\begin{array}{ccc}a+b& b+c& c+a\\ c+a& a+b& b+c\\ b+c& c+a& a+b\end{array}\right|=\left|\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right|+\left|\begin{array}{ccc}b& c& a\\ a& b& c\\ c& a& b\end{array}\right|=\left|\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right|+\left|\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right|=2\left|\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right|=-2\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$
Where $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is the required circulant matrix