Multiplying Matrices
Once you see a matrix as a transformation of space, matrix multiplication stops being a row-times-column ritual and becomes something simple: do one transformation, then another. The product matrix is the single transformation that has the same net effect.
Watch two transformations stack up below.
Multiplication is composition
Apply M₁ to space, then apply M₂ to the result. The combined map is written M₂M₁,
and it's itself a matrix — you can find it column by column: each column of the
product is where a basis vector ends up after both transformations. That's all the
row-by-column formula is really computing.
This is why you read matrix products right-to-left: in M₂M₁, the matrix nearest
the vector (M₁) acts first. It lines up with function composition, f(g(x)), where
g runs first.
Order matters
Because each matrix moves space in its own way, swapping the order usually changes
the result: M₁M₂ ≠ M₂M₁ in general. The animation shows it directly — shear-then-
rotate lands space somewhere different from rotate-then-shear. Matrix multiplication is
associative ((AB)C = A(BC)) but not commutative, and that single fact shapes
everything from 3-D graphics pipelines to the order you apply data transformations.
A useful sanity check rides along: the determinant multiplies, det(AB) = det A · det B. Areas scale by each step in turn, so the combined area-scale is just the product.
Sources
- Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press. — Matrix multiplication as composition.
- Lay, D. C., Lay, S. R., & McDonald, J. J. (2021). Linear Algebra and Its Applications (6th ed.). Pearson. — Properties of matrix multiplication (associative, non-commutative).