The determinant: how much a matrix stretches area
Slide 2's mirror kept every size but turned the plane over, and its determinant was −1. Now we look at that number on its own. Think of a photocopier set to 200%: every length doubles, so every shape on the page, big or small, gets 4 times the area. A matrix works the same way. It multiplies every area by one fixed number, its determinant, det A. You can read that number off the unit square, because the square lands on the slanted box made by the two columns. A minus sign means the plane was also flipped over, like a page turned face down. Zero means everything was squashed flat onto a line, and then nothing can undo the move.
Lesson 6 · slide 3Determinant- det A is the factor by which A multiplies every area (2D) or volume (3D).
- det A < 0: the matrix also flips space, like a mirror.
- det A = 0: space is squashed onto a line, a plane, or even a single point. Then A has no inverse.
- For 2 by 2, det A = ad − bc. Larger matrices follow from three rules (slide 4).
One stretch factor for every shape
A matrix may turn and slant shapes, but it multiplies every area by the same number. Take A = 3002. It makes every width 3 times bigger and every height 2 times bigger:
Read it off the unit square
A sends 10 to column 1 and 01 to column 2 (slide 1). So the unit square lands on the parallelogram (a slanted box) whose two sides are the columns. Its area is the stretch factor. Take A = 3112, with columns 31 and 12:
Why 5? Cut it out of cardboard. The parallelogram fits in a 4 by 3 box, area 12. Cut off the corners: two triangles of 3·1/2, two triangles of 1·2/2, and two small 1 by 1 squares.
The same cutting works for any columns ac and bd: the box is (a + b)(c + d), and taking off the corners leaves exactly ad − bc. So the number from lesson 3, the test for an inverse, is an area.
The sign: was the plane flipped?
Turn from column 1 to column 2 the short way. Counterclockwise, like 10 to 01? Then det A > 0. Clockwise? Then the plane was turned over, and det A < 0. Swap the two columns of the last matrix:
The mirror from slide 2, −1001, has det = −1: same area, flipped.
Zero: squashed flat
If both columns lie on one line, the parallelogram is flat, with area 0. Take A = 2110.5: column 1 is 21 and column 2 is 10.5, half of column 1.
Both 1−2 and 00 land on 00: 1−2 is in the nullspace (lesson 4). Many inputs give the same output, so nothing can undo A. The key fact: A is invertible exactly when det A ≠ 0. For 2 by 2 you saw this in lesson 3. It is true for every size, and slide 5 shows why.
Keep reading with the full course
Finish lesson 6 and you will be able to: see a matrix as a move of the plane, and build one from where two arrows land.
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Examples
Example 1 · A stretch
A = 2003.
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