Area Scaling: the Determinant

Lesson 10 of 12 · 7 min · Free, no account needed

What you will learn

The determinant of [[a, b], [c, d]] is ad - bc: how many times bigger every area gets.
det = 0 squashes the plane onto a line or a point. That cannot be undone.
A negative det mirrors the picture; its size is still the area factor.

A filter sends i hat to (3, 0) and j hat to (1, 2). How much bigger does every patch of the photo get?
Drag the lever to your bet.
Stretch the yellow patch with the two arrows.
The number inside is its new area. Reach each goal.
Swing j hat down onto i hat, and past it.
Watch the area and the blue flag.
That area factor is the determinant.
The determinant of [[a, b], [c, d]] is ad - bc. For our filter: 3 × 2 - 1 × 0 = 6.
det = 0: the plane is squashed flat.
det < 0: the picture is mirrored, and the area still grows by the size of det.
Game: area bingo. Hit each target area.
Every time you let go counts as a move.
Build a layer that squashes everything flat.
Four numbers, none of them zero, with determinant 0.
ad - bc = 0 exactly when the two columns lie on one line through (0, 0).
In AI, a layer that squashes the plane loses information for good.
Six inputs went through your layer. Tap the two inputs that landed on the same spot.
With det = 0 some different inputs land on the same output, so the layer cannot be undone.
What is the determinant of [[3, 1], [1, 2]]?
det = ad - bc = 3 × 2 - 1 × 1 = 5.
Which matrix mirrors the flag?
det [[0, 1], [1, 0]] = -1 < 0, so it mirrors. The others have det 2 and 1.
The determinant ad - bc says how much a matrix scales area.
Stamp each filter on your area meter.

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