How do you find the determinant of ((3, 4, 5, 2), (1, 0, 1, 0), (, 2, 3, 6, 3), (, 7, 2, 9, 4))?

May 3, 2016

$- 24$

Explanation:

$\left(\begin{matrix}\textcolor{t e a l}{3} & 4 & \textcolor{m a \ge n t a}{5} & 2 \\ \textcolor{t e a l}{1} & 0 & \textcolor{m a \ge n t a}{1} & 0 \\ \textcolor{t e a l}{2} & 3 & \textcolor{m a \ge n t a}{6} & 3 \\ \textcolor{t e a l}{7} & 2 & \textcolor{m a \ge n t a}{9} & 4\end{matrix}\right) =$
$\to$ column3 $-$ column1 $\to$ column3

$= \left(\begin{matrix}\textcolor{g o l d}{3} & 4 & 2 & 2 \\ \textcolor{b l u e}{1} & \textcolor{g o l d}{0} & \textcolor{g o l d}{0} & \textcolor{g o l d}{0} \\ \textcolor{g o l d}{2} & 3 & 4 & 3 \\ \textcolor{g o l d}{7} & 2 & 2 & 4\end{matrix}\right)$

$= 1 \cdot {\left(- 1\right)}^{2 + 1} \cdot \left(\begin{matrix}4 & 2 & 2 \\ 3 & 4 & 3 \\ 2 & 2 & 4\end{matrix}\right)$
$= - \left[64 + 12 + 12 - \left(16 + 24 + 24\right)\right]$
$= - 88 + 64 = - 24$