The transpose

The transpose A′ (also written Aᵀ) is made by turning each row of A into a column. If A = [aᵢⱼ] is m × n, then A′ = [aⱼᵢ] is n × m.

The entry in row i, column j of A′ is the entry in row j, column i of A. A row matrix transposes to a column matrix and the other way round.

(A′)′ = A: transposing twice returns the original.

(kA)′ = kA′ and (A + B)′ = A′ + B′.

(AB)′ = B′A′: the transpose of a product is the product of the transposes in reverse order. The orders show why: if A is m × n and B is n × p, then B′A′ is (p × n)(n × m), which is defined, while A′B′ generally is not.

Worked example: the column A with entries −2, 4, 5 times the row B = [1 3 −6] is the 3 × 3 matrix with rows (−2, −6, 12), (4, 12, −24), (5, 15, −30); B′A′ gives its transpose, confirming (AB)′ = B′A′.