The determinant is an important property of a square 🍱 Matrix. For a matrix , the determinant is

For higher dimensional matrices, we use Laplace expansion along a row or column. Along a column ,

where is a sub-matrix we obtain by deleting row and column .

Note that for an upper- or lower-triangular matrix, the determinant is the product of diagonal elements. This can be seen if we recursively choose to apply Laplace expansion along remaining columns or rows where only one element is nonzero.

Properties#

Determinants exhibit the following:

1. .
2. .
3. .
4. .
5. If and are similar (), .
6. Adding a multiple of a row to another does not change .
7. Multiplying a row by scales by .
8. Swapping two rows changes the sign of .

Furthermore, square matrix has if and only if the rank of is . In other words, has non-zero determinant if and only if it's invertible.