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

May 3, 2016

$434$

Explanation:

$\left(\begin{matrix}5 & \textcolor{t e a l}{3} & \textcolor{\mathmr{and} a n \ge}{1} & \textcolor{t u r q u o i s e}{2} \\ 0 & \textcolor{t e a l}{1} & \textcolor{\mathmr{and} a n \ge}{- 1} & \textcolor{t u r q u o i s e}{3} \\ 2 & \textcolor{t e a l}{7} & \textcolor{\mathmr{and} a n \ge}{- 4} & \textcolor{t u r q u o i s e}{1} \\ 3 & \textcolor{t e a l}{3} & \textcolor{\mathmr{and} a n \ge}{5} & \textcolor{t u r q u o i s e}{- 2}\end{matrix}\right) =$
$\to$ column3 $+$ column2 $\to$ column3
$\to$ column4 $- 3 \cdot$ column2 $\to$ column4

$= \left(\begin{matrix}5 & \textcolor{g o l d}{3} & 4 & - 7 \\ \textcolor{g o l d}{0} & \textcolor{b l u e}{1} & \textcolor{g o l d}{0} & \textcolor{g o l d}{0} \\ 2 & \textcolor{g o l d}{7} & 3 & - 20 \\ 3 & \textcolor{g o l d}{3} & 8 & - 11\end{matrix}\right)$

$= 1 \cdot {\left(- 1\right)}^{2 + 2} \cdot \left(\begin{matrix}5 & 4 & - 7 \\ 2 & 3 & - 20 \\ 3 & 8 & - 11\end{matrix}\right)$

$= - 165 - 240 - 112 - \left(- 63 - 800 - 88\right)$
$= - 517 + 951 = 434$