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Transpose of a product of matrices

Short answer

The transpose interchanges rows and columns. (AB)ᵀ entry equals (AB)Ɫ = Σ aᵢ bⱢ = Σ bⱢ aᵢ = (Bᵀ Aᵀ)Ɫ, so (AB)ᵀ = Bᵀ Aᵀ.

Define the transpose of a matrix. Prove that the transpose of the product of two matrices is the product of their transposes taken in reverse order.

The transpose of a matrix A of order m × n is the n × m matrix Aᵀ obtained by interchanging rows and columns; its (i, j) entry is aⱼᵢ.

Theorem

If A is m × n and B is n × p, then (AB)ᵀ = BᵀAᵀ.

Proof

Orders. AB is m × p, so (AB)ᵀ is p × m. Bᵀ is p × n and Aᵀ is n × m, so BᵀAᵀ is p × m. The orders agree.

Entries. Let A = [aᵢₖ] and B = [bₖⱼ]. The (j, i) entry of (AB)ᵀ is the (i, j) entry of AB:

[(AB)ᵀ]ⱼᵢ = (AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ

The (j, i) entry of BᵀAᵀ is

[BᵀAᵀ]ⱼᵢ = Σₖ (Bᵀ)ⱼₖ (Aᵀ)ₖᵢ = Σₖ bₖⱼ aᵢₖ

Since real or complex numbers commute under multiplication, the two sums are equal for every j and i. Hence (AB)ᵀ = BᵀAᵀ. ∎

Extension: (ABC)ᵀ = CᵀBᵀAᵀ, by applying the result twice.

Example. A = [1 2; 3 4], B = [0 1; 1 0]. AB = [2 1; 4 3], (AB)ᵀ = [2 4; 1 3]. Bᵀ Aᵀ = [0 1; 1 0][1 3; 2 4] = [2 4; 1 3] ✓.

✓ Answer checked by Claude Opus 5.5 (high effort)

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