Geometric Intuition
Geometrically, we can view determinant as the signed-volume scaling factor of a linear transformation
Recursive Definition
Definition
We can define determinant by *Laplace expansion:
where
Example
Definition Based on Properties
Denote the
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
is invertible iff- In the same vein, if
, its rows and columns must be linearly independent
- In the same vein, if
- If
is invertible, then