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

Feb 26, 2016

$- 100$

Explanation:

The easiest way to calculate this determinant is by eliminating the number 3 in column 3 row 4.
This can be done by multiplying column 4 by $\left(- \frac{3}{4}\right)$ and adding it to column 3 (remark that the value of the determinant remains the same)

$D e t . = \left[\begin{matrix}5 & 2 & 0 & 0 & - 2 \\ 0 & 1 & \left(4 - \frac{3}{4} \cdot 3\right) & 3 & 2 \\ 0 & 0 & \left(2 - \frac{3}{4} \cdot 6\right) & 6 & 3 \\ 0 & 0 & \left(3 - \frac{3}{\cancel{4}} \cdot \cancel{4}\right) & 4 & 1 \\ 0 & 0 & 0 & 0 & 2\end{matrix}\right]$

$= \left[\begin{matrix}5 & 2 & 0 & 0 & - 2 \\ 0 & 1 & \frac{7}{4} & 3 & 2 \\ 0 & 0 & - \frac{5}{2} & 6 & 3 \\ 0 & 0 & 0 & 4 & 1 \\ 0 & 0 & 0 & 0 & 2\end{matrix}\right]$

$= 5 \cdot 1 \cdot \left(- \frac{5}{\cancel{2}}\right) \cdot 4 \cdot \cancel{2} = - 100$