Matrix multiplication means doing transformations one after another. Inverses undo what a matrix does.
When you apply transformation A and then transformation B, the combined effect is a single transformation. This combined transformation is represented by the matrix product BA (note: B times A, not A times B).
If a transformation can be "undone," its inverse exists. But not all transformations can be reversed—some destroy information by collapsing space.
Watch as we apply rotation A, then shear B. The final result is exactly what the product matrix BA produces in a single step.
Key insight: Applying A then B is the same as applying the single matrix BA (note the order!). Matrix multiplication BA means "do A first, then B".
Matrix multiplication is not commutative. Rotating then scaling gives a different result than scaling then rotating.
AB ≠ BA (order matters!)
Rotating then scaling gives a different result than scaling then rotating.
The inverse matrix A⁻¹ reverses what A does. Apply A, then A⁻¹, and you're back where you started. The identity matrix I (which does nothing) satisfies A·A⁻¹ = I.
The inverse matrix A⁻¹ undoes what A does. Applying A then A⁻¹ (or vice versa) returns to the identity—no transformation at all.
A singular matrix (determinant = 0) collapses space to a lower dimension. Once collapsed, you can't reconstruct where points came from—no inverse exists.
As the matrix approaches singular (det → 0), space collapses to a lower dimension. Information is lost—you can't tell where points came from, so there's no inverse.
Next: Let's explore determinants—the number that tells us how much space stretches or compresses.