Question Number 204041 by hardmath last updated on 04/Feb/24
$$\mathrm{Find}:\:\:\:\begin{vmatrix}{\mathrm{1}}&{\mathrm{7}}&{−\mathrm{1}}\\{\mathrm{9}}&{−\mathrm{3}}&{\mathrm{5}}\\{−\mathrm{1}}&{\mathrm{5}}&{\mathrm{3}}\end{vmatrix}=\:? \\ $$
Answered by AST last updated on 04/Feb/24
$$=\mathrm{1}\begin{vmatrix}{−\mathrm{3}}&{\mathrm{5}}\\{\mathrm{5}}&{\mathrm{3}}\end{vmatrix}−\mathrm{7}\begin{vmatrix}{\mathrm{9}}&{\mathrm{5}}\\{−\mathrm{1}}&{\mathrm{3}}\end{vmatrix}−\mathrm{1}\begin{vmatrix}{\mathrm{9}}&{−\mathrm{3}}\\{−\mathrm{1}}&{\mathrm{5}}\end{vmatrix} \\ $$$$=−\mathrm{9}−\mathrm{25}−\mathrm{7}\left(\mathrm{27}+\mathrm{5}\right)−\mathrm{1}\left(\mathrm{45}−\mathrm{3}\right)=−\mathrm{300} \\ $$
Answered by som(math1967) last updated on 05/Feb/24
$${C}_{\mathrm{2}} \rightarrow{C}_{\mathrm{2}} −\mathrm{7}{C}_{\mathrm{1}} \\ $$$${C}_{\mathrm{3}} \rightarrow{C}_{\mathrm{3}} +{C}_{\mathrm{1}} \\ $$$$\begin{vmatrix}{\mathrm{1}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{9}}&{−\mathrm{66}}&{\mathrm{14}}\\{−\mathrm{1}}&{\mathrm{12}}&{\mathrm{2}}\end{vmatrix} \\ $$$$=−\mathrm{66}×\left(\mathrm{2}\right)−\mathrm{12}×\mathrm{14} \\ $$$$=−\mathrm{132}−\mathrm{168}=−\mathrm{300} \\ $$