Matrices
Matrices and determinants
- A matrix is a block of numbers; multiply row × column. The identity $I$ leaves a matrix unchanged.
- For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$: the determinant is $\det A = ad - bc$.
- If $\det A \neq 0$ (non-singular), the inverse is $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
Practice
What is the determinant of the matrix [[3, 1], [2, 4]] (ad − bc)?
det = 3×4 − 1×2 = 12 − 2 = 10.
Practice
A 2×2 matrix has an inverse exactly when its determinant is:
A non-singular matrix (det ≠ 0) is invertible; det = 0 means singular (no inverse).
Transformations
- A $2\times2$ matrix represents a transformation of the plane; $\det A$ = the area scale factor.
- A product $AB$ means "do $B$, then $A$". Points/lines that don't move are invariant.
Practice
For a transformation matrix, the determinant gives the:
The determinant is the factor by which areas are scaled by the transformation.
You've got it
Key idea
- $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$; invertible when $\det \neq 0$
- $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
- a matrix is a transformation; $\det$ = area scale factor; $AB$ = "do $B$, then $A$"