Question

Given that $\left|\begin{array}{ccc}a& d& 1\\ b& e& 1\\ c& f& 1\end{array}\right|=-5,\left|\begin{array}{ccc}a& b& c\\ 1& 2& 3\\ d& e& f\end{array}\right|=-3$, then the value of $\left|\begin{array}{ccc}c& 3& f\\ b& 1& e\\ a& -1& d\end{array}\right|$ is

Answer 1

$2\left|\begin{array}{ccc}a& d& 1\\ b& e& 1\\ c& f& 3\end{array}\right|-3\left|\begin{array}{ccc}a& d& 1\\ b& e& 1\\ c& f& 1\end{array}\right|=21$

$\Rightarrow \left|\begin{array}{ccc}a& d& -1\\ b& e& 1\\ c& f& 3\end{array}\right|=21$
$R_1\leftrightarrow R_3$
$\Rightarrow \left|\begin{array}{ccc}c& f& 3\\ b& e& 1\\ a& d& -1\end{array}\right|=-21$
$C_2\leftrightarrow C_3$ we get determinant.
$\left|\begin{array}{ccc}c& 3& f\\ b& 1& e\\ a& -1& d\end{array}\right|=21$