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

May 3, 2016

$255$

#### Explanation:

$\left(\begin{matrix}\textcolor{t e a l}{5} & \textcolor{t e a l}{6} & \textcolor{t e a l}{0} & \textcolor{t e a l}{0} \\ 0 & 1 & - 1 & 2 \\ - 3 & 4 & - 5 & 1 \\ \textcolor{\mathmr{and} a n \ge}{1} & \textcolor{\mathmr{and} a n \ge}{6} & \textcolor{\mathmr{and} a n \ge}{0} & \textcolor{\mathmr{and} a n \ge}{3}\end{matrix}\right) =$
$\to$ row4 $-$ row1 $\to$ row4

$= \left(\begin{matrix}\textcolor{b l u e}{5} & \textcolor{c r i m s o n}{6} & 0 & 0 \\ 0 & 1 & - 1 & 2 \\ - 3 & 4 & - 5 & 1 \\ - 4 & 0 & 0 & 3\end{matrix}\right)$

$= 5 \cdot {\left(- 1\right)}^{1 + 1} \left(\begin{matrix}1 & - 1 & 2 \\ 4 & - 5 & 1 \\ 0 & 0 & 3\end{matrix}\right) + 6 \cdot {\left(- 1\right)}^{1 + 2} \cdot \left(\begin{matrix}0 & - 1 & 2 \\ - 3 & - 5 & 1 \\ - 4 & 0 & 3\end{matrix}\right)$

$= 5 \cdot \left[- 15 - \left(- 12\right)\right] - 6 \cdot \left[4 - \left(40 + 9\right)\right]$
$= 5 \cdot \left[- 3\right] - 6 \cdot \left[- 45\right] = - 15 + 270 = 255$