When a Transform Flattens Your Sprite

Game engines move, scale and skew sprites with 2×2 matrices. The determinant says how the matrix changes area: double it, halve it, flip it (negative) or crush it flat (zero). A flattened sprite can't be un-flattened, which is exactly when the matrix has no inverse.

How it works

Determinants and invertibility

In one sentence

For $A = \begin{bmatrix} a & b \\ c & d\end{bmatrix}$ , $\det A = ad - bc$ ; the image of a region under $A$ has $|\det A|$ times its area, $\det(AB) = \det A\det B$ , and $A$ is invertible exactly when $\det A \ne 0$ (the Invertible Matrix Theorem).

General formula

{\htmlData{sym=0}{A}} = \begin{bmatrix} {\htmlData{sym=1}{a}} & {\htmlData{sym=2}{b}} \\ {\htmlData{sym=3}{c}} & {\htmlData{sym=4}{d}} \end{bmatrix}, \qquad \text{area scale} = |\det {\htmlData{sym=0}{A}}| = |{\htmlData{sym=1}{a}}{\htmlData{sym=4}{d}} - {\htmlData{sym=2}{b}}{\htmlData{sym=3}{c}}|, \qquad {\htmlData{sym=0}{A}} \text{ invertible} \iff \det {\htmlData{sym=0}{A}} \ne 0

The determinant is the signed area of the parallelogram the columns span: how much the map stretches area, and whether it flips it over.

What the symbols mean

$A$
Transform matrix
Applied to every point of the sprite.
$a$
Top-left entry
x-stretch: where (1, 0) lands horizontally.
$b$
Top-right entry
How far (0, 1) is pushed sideways: shear.
$c$
Bottom-left entry
How far (1, 0) is pushed up.
$d$
Bottom-right entry
y-stretch: where (0, 1) lands vertically.

The columns of $A$ are where the unit vectors land: $(a, c)$ and $(b, d)$ . The unit square becomes the parallelogram they span.
That parallelogram's area is $|ad - bc|$ , so every shape's area is multiplied by $|\det A|$ .
A negative determinant means the sprite is mirrored: its orientation flips.
At $\det A = 0$ the parallelogram collapses to a line. Many points land on the same spot, so nothing can undo it: no inverse.

Learn the math

Linear TransformationsLin Alg · Lesson 7, Linear Equations in Linear Algebra The Invertible Matrix TheoremLin Alg · Lesson 11, Matrix Algebra Introduction to DeterminantsLin Alg · Lesson 16, Determinants Properties of DeterminantsLin Alg · Lesson 17, Determinants Cramer's Rule, Volume and TransformationsLin Alg · Lesson 18, Determinants

Learn the math

Linear TransformationsLin Alg · Lesson 7, Linear Equations in Linear Algebra

The Invertible Matrix TheoremLin Alg · Lesson 11, Matrix Algebra

Introduction to DeterminantsLin Alg · Lesson 16, Determinants

Properties of DeterminantsLin Alg · Lesson 17, Determinants

Cramer's Rule, Volume and TransformationsLin Alg · Lesson 18, Determinants