Matrix multiplication as composition — 3Blue1Brown · Essence of Linear Algebra Ch.4
See matrix multiplication as applying one transformation after another.
Step 1 (right matrix, applied first):
Step 2 (left matrix, applied second):
M₁ =
M₂ =
M₂·M₁ =
Composition — Step 1 (orange, M₁) applies first, step 2 (purple, M₂) applies second. The result equals applying the single green composition matrix M₂·M₁ directly. Press play to watch the two stages.
Shear then rotate (M₂·M₁)
Rotate then shear (M₁·M₂)
Order matters — Shear then rotate gives a different final grid than rotate then shear. Watch both panels animate — visual proof matrix multiplication is not commutative.
î and ĵ arrows show where the basis vectors land at each stage