Question

If $xyz=-2007$ and $\Delta =0$, where
$\Delta =\left|\begin{array}{ccc}a+x& b& c\\ a& b+y& c\\ a& b& c+z\end{array}\right|$
Find the value of $ayz+bzx+cxy.$

• A. $2007$
• B. $-2007$
• C. $0$
• D. None of the above

Answer 1

The correct option is A $2007$

$\Delta =\left|\begin{array}{ccc}a+x& b& c\\ a& b+y& c\\ a& b& c+z\end{array}\right|$
$C_1\to C_1+C_2+C_3$
$\Delta =\left|\begin{array}{ccc}a+b+c+x& b& c\\ a+b+c+y& b+y& c\\ a+b+c+z& b& c+z\end{array}\right|$
$\Delta =\left|\begin{array}{ccc}a+b+c& b& c\\ a+b+c& b+y& c\\ a+b+c& b& c+z\end{array}\right|+\left|\begin{array}{ccc}x& b& c\\ y& b+y& c\\ z& b& c+z\end{array}\right|$
$\Delta =(a+b+c)\left|\begin{array}{ccc}1& b& c\\ 1& b+y& c\\ 1& b& c+z\end{array}\right|+\left|\begin{array}{ccc}x& b& c\\ y& b+y& c\\ z& b& c+z\end{array}\right|$
Applying in the first determinant $R_2\to R_2-R_1,R_3\to R_3-R_1$
$\Delta =(a+b+c)\left|\begin{array}{ccc}1& b& c\\ 0& y& 0\\ 0& 0& z\end{array}\right|+\left|\begin{array}{ccc}x& b& c\\ y& b+y& c\\ z& b& c+z\end{array}\right|$
$=ayz+byz+cyz+x(bc+bz+cy+yz-bc)-b(cy+yz-cz)+c(by-bz-yz)$

$\Delta =ayz+bxz+cxy+xyz$

$\Delta =ayz+bxz+cxy-2007$