Transpose of a Matrix

The transpose of a matrix is one of the most fundamental operations in linear algebra. Given a matrix A, its transpose — written as Aᵀ (read 'A transpose') — is formed by flipping the matrix over its main diagonal, turning every row into a column and every column into a row. If A has dimensions m × n (m rows and n columns), then Aᵀ has dimensions n × m. Transposition appears throughout mathematics, physics, engineering, and computer science. It is essential in defining symmetric matrices, computing dot products using matrix notation, solving systems of linear equations via the normal equations, and working with transformations in geometry. Understanding the transpose and its properties is a prerequisite for studying orthogonality, least-squares methods, and eigenvalue problems. The operation is surprisingly powerful despite its simplicity. Several elegant properties — such as (AB)ᵀ = BᵀAᵀ and (Aᵀ)ᵀ = A — make the transpose a reliable tool for algebraic manipulation. Learning these properties allows students to simplify complex matrix expressions and recognize special structure in matrices that arise in real-world applications.

Key takeaways
  • Identify the dimensions of the original matrix
  • Apply the transpose rule entry by entry
  • Check the main diagonal
  • Learn the key algebraic properties
  • Recognize symmetric and skew-symmetric matrices

Identify the dimensions of the original matrix

Start by noting that matrix A has m rows and n columns, written as m × n. The entry in row i and column j is written aᵢⱼ. For example, a 2 × 3 matrix has 2 rows and 3 columns, giving 6 entries total.

Apply the transpose rule entry by entry

The defining rule is: the (i, j) entry of Aᵀ equals the (j, i) entry of A. In symbols, (Aᵀ)ᵢⱼ = aⱼᵢ. Every row of A becomes a column of Aᵀ, and every column of A becomes a row of Aᵀ. The result has dimensions n × m.

Check the main diagonal

The main diagonal entries (where i = j) are unchanged by transposition: aᵢᵢ stays in place. This is because swapping row and column indices on the diagonal gives the same position. For a square matrix, only the off-diagonal entries move.

Learn the key algebraic properties

Four properties govern the transpose: (1) (Aᵀ)ᵀ = A — applying transpose twice returns the original matrix. (2) (A + B)ᵀ = Aᵀ + Bᵀ — transpose distributes over addition. (3) (cA)ᵀ = cAᵀ for any scalar c — scalars factor out. (4) (AB)ᵀ = BᵀAᵀ — the order reverses when transposing a product, exactly like the reversal rule for inverses.

Recognize symmetric and skew-symmetric matrices

A square matrix is called symmetric if Aᵀ = A, meaning aᵢⱼ = aⱼᵢ for all i, j. It is called skew-symmetric (or anti-symmetric) if Aᵀ = −A, meaning aᵢⱼ = −aⱼᵢ and all diagonal entries are zero. Any square matrix can be written as the sum of a symmetric and a skew-symmetric part: A = ½(A + Aᵀ) + ½(A − Aᵀ).

The Key Equation

(A⊤)ij=aji,A∈Rm×n⇒A⊤∈Rn×m(A^\top)_{ij} = a_{ji}, \quad A \in \mathbb{R}^{m \times n} \Rightarrow A^\top \in \mathbb{R}^{n \times m}
  • A — Original matrix (dimensionless): The m × n matrix being transposed, with entries aᵢⱼ.
  • A^\top — Transpose of A (dimensionless): The resulting n × m matrix whose (i,j) entry is the (j,i) entry of A.
  • a_{ji} — Entry of A at row j, column i (dimensionless): The specific numerical element of A located in row j and column i, which becomes the (i,j) entry of Aᵀ.
  • m, n — Matrix dimensions (dimensionless): m is the number of rows and n is the number of columns of the original matrix A.

Worked Example

Problem. Let A = [[1, 2, 3], [4, 5, 6]]. (a) Find Aᵀ. (b) Verify that (Aᵀ)ᵀ = A. (c) Let B = [[1, 0], [0, 2], [1, 1]]. Compute (AB)ᵀ and BᵀAᵀ, then verify they are equal.

A⊤=(142536),(AB)⊤=B⊤A⊤=(410716)A^\top = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}, \quad (AB)^\top = B^\top A^\top = \begin{pmatrix} 4 & 10 \\ 7 & 16 \end{pmatrix}

Answer. Aᵀ = [[1,4],[2,5],[3,6]]; (AB)ᵀ = BᵀAᵀ = [[4,10],[7,16]]

Figure
Diagram
Matrix A (2×3)Aᵀ (3×2)123456r1r2c1c2c3142536c1c2r1r2r3Transposerows → columnsRows of A become columns of Aᵀ; columns become rows.Matching colors show where each entry moves.
Origins · 1 minArthur Cayley and how this was discovered
AC
Arthur Cayley
1821–1895 · British

Arthur Cayley was the first mathematician to formally develop matrix algebra as a systematic mathematical structure. In his landmark 1858 paper 'A Memoir on the Theory of Matrices,' he introduced matrix multiplication, the identity matrix, and the inverse of a matrix, laying the groundwork for all modern linear algebra including the systematic study of the transpose operation. Cayley was trained as a lawyer and worked as a solicitor for 14 years before becoming a mathematics professor at Cambridge — yet during that legal career he still managed to publish nearly 300 mathematical papers.

  • Matrix Multiplication — The reversal rule (AB)ᵀ = BᵀAᵀ directly connects the transpose to matrix multiplication, and understanding both is essential for manipulating matrix expressions.
  • Symmetric Matrices — A symmetric matrix is defined by the condition Aᵀ = A, making the transpose the core concept needed to identify and work with symmetric matrices.
  • Inverse of a Matrix — The transpose and inverse share the reversal-of-order property under products: (AB)⁻¹ = B⁻¹A⁻¹, and for orthogonal matrices the transpose equals the inverse.
  • Dot Product — The dot product of two vectors u and v can be written as uᵀv using the transpose, bridging vector operations and matrix algebra.