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

Feb 14, 2016

See explanation...

#### Explanation:

If you swap two rows or columns the deteminant is negated.

If you add or subtract any multiple of one row from another, the determinant is unchanged.

If the only non-zero value in the first row or column is in the top left corner then the determinant is that value multiplied by the submatrix formed by the other rows and columns.

So we can simplify as follows:

$\left\mid \begin{matrix}1 & 2 & 5 & 0 & 1 \\ 2 & 3 & 7 & 1 & 9 \\ 1 & 2 & 3 & 0 & 3 \\ 0 & 0 & 1 & 0 & 0 \\ 3 & 2 & - 4 & 0 & 1\end{matrix} \right\mid$

=-abs((0,0,1,0,0),(2,3,7,1,9),(1,2,3,0,3),(1,2,5,0,1),(3,2,-4,0,1)) color(grey)(" " { "rows 1 and 4 swapped" })

=abs((1,0,0,0,0),(7,3,2,1,9),(3,2,1,0,3),(5,2,1,0,1),(-4,2,3,0,1)) color(grey)(" " { "columns 1 and 3 swapped" })

$= \left\mid \begin{matrix}3 & 2 & 1 & 9 \\ 2 & 1 & 0 & 3 \\ 2 & 1 & 0 & 1 \\ 2 & 3 & 0 & 1\end{matrix} \right\mid$

=-abs((1,2,3,9),(0,1,2,3),(0,1,2,1),(0,3,2,1)) color(grey)(" " { "columns 1 and 3 swapped" })

$= - \left\mid \begin{matrix}1 & 2 & 3 \\ 1 & 2 & 1 \\ 3 & 2 & 1\end{matrix} \right\mid$

=-abs((0,0,2),(1,2,1),(3,2,1)) color(grey)(" " { "subtracted row 2 from row 1" })

=abs((2,0,0),(1,2,1),(1,2,3)) color(grey)(" " { "columns 1 and 3 swapped" })

$= 2 \left\mid \begin{matrix}2 & 1 \\ 2 & 3\end{matrix} \right\mid = 2 \left(2 \cdot 3 - 1 \cdot 2\right) = 8$