What a matrix does to the plane

See a 2×2 matrix as a recipe for moving i-hat and j-hat, and watch determinants and eigenvectors appear. About 9 minutes, A-level Further Maths.

What you will do

1. The plane. This is the plane with its grid, a unit square and a little house. A matrix is a rule that moves every point. We will see that the whole rule is decided by just two arrows: î = (1, 0) and ĵ = (0, 1).
2. The columns are the landing spots. The first column of the matrix is where î lands; the second column is where ĵ lands. Right now the matrix is the identity: nothing moves. Drag the red î arrow tip to (2, 1) and watch the grid follow.
3. How areas change. Now the matrix is pmatrix2&0\0&3pmatrix: x is stretched by 2 and y by 3. The unit square becomes a 2 by 3 rectangle.
4. A rotation. A 90° rotation sends î to (0, 1) and ĵ to (-1, 0). Shape and size are unchanged, so the determinant is det.
5. A shear and its eigenvector. A shear slides each row sideways. Areas are still preserved (determinant det) but shapes lean. Most directions get knocked off their line; the x-axis direction is special — it stays put. That is an eigenvector, drawn in the plane.
6. Flip the plane. A negative determinant means the plane is turned over, like a reflection. Drag î and ĵ (or type entries) so that the determinant goes negative and see the colour change.
7. Squash it flat. What if both arrows point the same way? The whole plane collapses onto a line: area 0. Make the determinant almost zero.
8. No way back. A flattened plane cannot be un-flattened: this matrix is singular and has no inverse. The panel says so. Any matrix with non-zero determinant can be undone by its inverse.
9. One matrix after another. Apply A (rotate 90°) and then B (scale by 2). The product BA does both at once, and areas multiply: the determinants multiply too.
10. Your turn. Everything is unlocked. Try the presets, type entries like cos(pi/6) or 1/2, drag the arrows, and switch composition on to build your own map.