Question

The $\left|\begin{array}{ccc}b+c& c+a& a+b\\ q+r& r+p& p+q\\ y+z& z+x& x+y\end{array}\right|$ can be simplified to

• A. 2$\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$
• B. $\left|\begin{array}{ccc}p& q& r\\ a& b& c\\ x& y& z\end{array}\right|$
• C. $\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$
• D. $\left|\begin{array}{ccc}p& a& b\\ x& q& c\\ y& z& r\end{array}\right|$

Answer 1

The correct option is A 2$\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$

$=$$\left|\begin{array}{ccc}b+c& c+a& a+b\\ q+r& r+p& p+q\\ y+z& z+x& x+y\end{array}\right|$

$=$$\left|\begin{array}{ccc}b& c& a\\ q& r& p\\ y& z& x\end{array}\right|$ $+$ $\left|\begin{array}{ccc}c& a& b\\ r& p& q\\ z& x& y\end{array}\right|$

We could also write it as, because we could shift matrix in columns:

$=$ $\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$ $+$ $\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$

$=$ 2 $\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$