📖 Determinant

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 .

Info

For an upper-triangular matrix, the determinant is the product of diagonal elements.

Properties §

Determinants exhibit the following.

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