Determinant

Determinant is based on Transformations, and can therefore be extended to matrices

Determinants help us to quantify how much Transformations scale unit areas of a metric space - meaning in a 2 dimensional space with $i$ and $j$ unit vectors, meaning a 1x1 = 1 area, if our Transformation $T$ turns that into a 2x3 area then our determinant would be 6

Determinants for 3 dimensional spaces would correspond to Volume of a cube, whereas 2 dimensional would be on flat area

Determinants also have signs for when there is an orientation flip, and a determinant of 0 would correspond to reduction of dimensions - meaning 2d down to 1d or single point

Given a $2 \times 2$ matrix:

$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

The determinant of $A$ is:

$$\det(A) = ad - bc$$

Theorems

Commutative Product

The commutativity makes sense because it would just be 2 transformations chained onto each other...most linear transformations are just linear chains of transformations, so having the determinant also be one isn't surprising

$det(A) \times det(B) = det(AB)$

Problems