Question

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

Answer 1

`|(1, -2, 3, 2), (2, -1, 0, 4), (-3, 4, 0, -2), (-1, 1, 1, 5)| = 81`

Explanation:

• You can add or subtract multiples of any row or column to any other row or column without altering the determinant.

• You can swap two rows or columns, resulting in inverting the sign of the determinant.

• You can cyclically permute three rows or columns to leave the determinant unchanged (since this is equivalent to two swaps).

• The determinant of an upper or lower triangular matrix is the product of the diagonal.

So:

`|(1, -2, 3, 2), (2, -1, 0, 4), (-3, 4, 0, -2), (-1, 1, 1, 5)|" "` subtract `3` times row `4` from row `1` row to get:

`= |(4, -5, 0, -13), (2, -1, 0, 4), (-3, 4, 0, -2), (-1, 1, 1, 5)|" "` swap columns `1` and `3` to get:

`= -|(0, -5, 4, -13), (0, -1, 2, 4), (0, 4, -3, -2), (1, 1, -1, 5)|" "` swap rows `1` and `4` to get:

`= |(1, 1, -1, 5), (0, -1, 2, 4), (0, 4, -3, -2), (0, -5, 4, -13)|" "` add `4` times row `2` to row `3` to get:

`= |(1, 1, -1, 5), (0, -1, 2, 4), (0, 0, 5, 14), (0, -5, 4, -13)|" "` subtract `5` times row `2` from row `4` to get:

`= |(1, 1, -1, 5), (0, -1, 2, 4), (0, 0, 5, 14), (0, 0, -6, -33)|" "` add `6/5` times row `3` to row `4` to get:

`= |(1, 1, -1, 5), (0, -1, 2, 4), (0, 0, 5, 14), (0, 0, 0, -81/5)|`

`=1 * (-1) * 5 * (-81/5) = 81`