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More on matrix multiplication

Associativity, composition, transpose and consequences

Why does matrix multiplication represent composition?

Matrix multiplication represents composition because AB applies B first and then applies A to the result. The identity ( AB )c = A( B c ) shows that the single matrix AB produces exactly the same output as the two transformations performed in sequence.

Open chapter PDF Key ideas Chapter index Back to Linear Algebra

Key ideas

This chapter takes the basic definition of matrix multiplication and develops its main structural consequences.

  • Associativity follows by applying the vector identity ( A B ) c = A ( B c ) to each column of a matrix
  • The same column-wise definition also gives distributivity, scaling rules and the identity matrix
  • The product AB represents the composition “apply B, then apply A”
  • The transpose reverses multiplication order: (A B)ᵀ = Bᵀ Aᵀ
  • The adjoint identity ( A v ) · w = v · ( Aᵀ w ) links transpose to dot product and projection

The chapter uses these consequences to connect algebraic rules, composition of maps and geometric interpretation.

Why is matrix multiplication associative?

It is associative because both ( A B ) C and A ( B C ) act the same way on every column of C. Once we know that ( A B ) c = A ( B c ) for every vector c, the equality extends column by column to any matrix C.

Related chapters

  • Matrix multiplication introduces the core definitions and the column-wise, entry-wise, row-wise and outer-product viewpoints used here
  • Linear transformations gives the language of composition that explains why AB means apply B then A
  • Projection supports the dot-product and adjoint viewpoint developed through transpose
  • The four subspaces develops the rank, null-space and row/column-space consequences mentioned at the end of this chapter

Chapter contents

The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.

Topic Pages
Associativity of matrix multiplication: (AB)C = A(BC) 1–2
Other algebraic properties of matrix multiplication 2–4
Matrix multiplication as composition of linear transformations 4–7
Composition of linear transformations vs composition of algebraic functions 7
Transpose of matrix product (AB)T 8–9
Transpose and dot product 10–11
Matrix multiplication: what was defined and what followed 12–13
Open PDF

Why does transposing a product reverse the order?

The transpose swaps rows with columns, so the row–column pairing in AB becomes the row–column pairing in Bᵀ Aᵀ. That is why (A B)ᵀ equals Bᵀ Aᵀ rather than Aᵀ Bᵀ.

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