Geometric Intuition

Geometrically, we can view determinant as the signed-volume scaling factor of a linear transformation determinant_volume_change.png

Recursive Definition

Definition

We can define determinant by *Laplace expansion:

where is element at the th row and th column of , and is the determinant of the submatrix obtained by removing the th row and th columns from B.

Definition Based on Properties

Denote the matrix as , where denotes the th column:

Definition

The determinant of is defined as the unique real-valued function

  • for

In order to be able to use this definition, we can prove that such a function that satisfying all the above conditions exists and is unique.

Some Other Theorems