# How do you find the determinant of ((2, 11, -3, 1), (1, 5, 7, -4), (6, 13, -5, 2), (4, 22, -6, 2))?

May 3, 2016

$0$

#### Explanation:

$\left(\begin{matrix}\textcolor{b l u e}{2} & \textcolor{b l u e}{11} & \textcolor{b l u e}{- 3} & \textcolor{b l u e}{1} \\ 1 & 5 & 7 & - 4 \\ 6 & 13 & - 5 & 2 \\ \textcolor{c r i m s o n}{4} & \textcolor{c r i m s o n}{22} & \textcolor{c r i m s o n}{- 6} & \textcolor{c r i m s o n}{2}\end{matrix}\right) = 2 \cdot \left(\begin{matrix}\textcolor{b l u e}{2} & \textcolor{b l u e}{11} & \textcolor{b l u e}{- 3} & \textcolor{b l u e}{1} \\ 1 & 5 & 7 & - 4 \\ 6 & 13 & - 5 & 2 \\ \textcolor{c r i m s o n}{2} & \textcolor{c r i m s o n}{11} & \textcolor{c r i m s o n}{- 3} & \textcolor{c r i m s o n}{1}\end{matrix}\right)$

When a determinant has two lines or 2 columns equal (or what leads to the same conclusion, when it has two lines or 2 columns proportional), this determinant is equal to zero.

Therefore, since, in the present case, row1 $=$ row4 :
$D e t . = 0$