Matrix Transpose Calculator
Enter your matrix entries separated by spaces (columns) and semicolons (rows), choose the number of rows and columns, and get the transposed matrix instantly. The calculator shows the full step-by-step working, detects structural properties like symmetry, and explains what each step means. Supports integers, decimals, and fractions such as 3/4.
What is a matrix transpose?
The transpose of a matrix A, written A^T (or A'), is formed by flipping the matrix over its main diagonal so that each row of A becomes a column of A^T and each column becomes a row. Formally, if A is an m x n matrix with entry a_ij at row i and column j, then A^T is an n x m matrix whose entry at row j and column i equals a_ij. This simple swap has deep consequences throughout linear algebra, statistics, and engineering. Every symmetric matrix used in covariance estimation or physics is defined by the condition A = A^T. Every orthogonal rotation matrix satisfies A^T A = I, meaning transpose acts as the matrix inverse.
How to use this calculator
Select the number of rows and columns from the dropdown menus, then type your matrix entries in the text field. Use a space or comma to separate values within a row, and a semicolon to move to the next row. For example, a 2 x 3 matrix with values 1 2 3 in the first row and 4 5 6 in the second would be entered as "1 2 3; 4 5 6". Fractions such as 1/2 or 3/4 are accepted. The calculator instantly shows the transposed matrix, its dimensions, the trace for square matrices, and whether the matrix is symmetric or skew-symmetric. The "Show your work" panel walks through every index mapping step by step.
Transpose properties every student should know
Four identities come up constantly. First, double transposition recovers the original: (A^T)^T = A. Second, the transpose of a sum is the sum of the transposes: (A + B)^T = A^T + B^T. Third, the product rule reverses order: (AB)^T = B^T A^T, a fact responsible for many identities in multivariate statistics and signal processing. Fourth, for an invertible matrix, transposing and inverting commute: (A^-1)^T = (A^T)^-1, so both operations together return the same result regardless of which is applied first. The trace identity tr(A^T) = tr(A) means the diagonal sum is a structural invariant of a matrix, unchanged by transposition.
Symmetric and skew-symmetric matrices
A square matrix A is symmetric when A = A^T, meaning the entry in row i column j equals the entry in row j column i for every pair. Covariance matrices, Hessians, and Gram matrices are always symmetric. A square matrix is skew-symmetric when A = -A^T, which forces every diagonal entry to be zero because a_ii = -a_ii implies a_ii = 0. Every square matrix can be written as the sum of one symmetric and one skew-symmetric part: A = (A + A^T)/2 + (A - A^T)/2. Symmetric matrices have the useful property that all their eigenvalues are real (the spectral theorem), which is why they are central to principal component analysis, quadratic forms, and physical models where real-valued energy levels are required.
Applications of matrix transposition
Transposition appears wherever you convert between row and column representations of data. In machine learning, the normal equation for ordinary least squares is solved using X^T X, the Gram matrix. In physics, the transpose of the stress tensor relates to the moment of a force. In computer graphics, transposing a transformation matrix converts between row-major and column-major conventions used by different graphics APIs. In signal processing, the matched filter and the correlation matrix both rely on conjugate transposes. For real-valued matrices, the conjugate transpose (also called the Hermitian transpose or Hermitian adjoint, denoted A*) equals the ordinary transpose because complex conjugation has no effect on real numbers.
Key properties of matrix transposition
| Property | Identity | What it means |
|---|---|---|
| Double transpose | (A^T)^T = A | Transposing twice gives back the original |
| Scalar multiple | (cA)^T = c A^T | Scalars factor out of the transpose |
| Sum | (A + B)^T = A^T + B^T | Transpose distributes over addition |
| Product | (AB)^T = B^T A^T | Order reverses when transposing a product |
| Determinant | det(A^T) = det(A) | Transpose preserves the determinant |
| Inverse | (A^-1)^T = (A^T)^-1 | Transpose and inverse commute |
| Trace | tr(A^T) = tr(A) | Diagonal sum is unchanged by transposition |
| Symmetric | A^T = A | A matrix that equals its own transpose |
| Skew-symmetric | A^T = -A | A matrix that equals the negative of its transpose |
| Orthogonal | A^T = A^-1 | Columns/rows form orthonormal sets |
These identities hold for all conformable matrices A, B and scalar c.
Frequently asked questions
What does transposing a matrix do?
Transposing a matrix swaps its rows and columns. The entry at row i, column j in the original matrix moves to row j, column i in the transposed matrix. If the original has m rows and n columns, the transpose has n rows and m columns. The total number of entries stays the same, but the arrangement changes.
Why does the product rule for transposes reverse the order?
The identity (AB)^T = B^T A^T follows from dimension compatibility. If A is m x n and B is n x p, then AB is m x p, and (AB)^T must be p x m. B^T is p x n and A^T is n x m, so B^T A^T is p x m, which matches. Applying the transposes in the same order (A^T B^T) would produce an n x m times p x n product, which is not conformable for most dimensions.
What is the difference between the transpose and the Hermitian (conjugate) transpose?
The ordinary transpose A^T swaps rows and columns without changing the values of the entries. The Hermitian transpose (or conjugate transpose) A* swaps rows and columns and then takes the complex conjugate of every entry, replacing each a + bi with a - bi. For matrices with real-valued entries, the two operations are identical because the complex conjugate of a real number is itself. The Hermitian transpose is the natural generalization to complex matrices and appears in quantum mechanics and complex signal processing.
How can I tell if a matrix is symmetric without transposing it?
A square matrix is symmetric if and only if the entry in row i, column j equals the entry in row j, column i for every pair of indices. In practice, you check whether the matrix is mirror-symmetric about its main diagonal. If any off-diagonal pair a_ij and a_ji differs, the matrix is not symmetric. Alternatively, subtract A^T from A: if the result is the zero matrix, A is symmetric.
Does transposing a matrix change its determinant?
No. The determinant of a matrix equals the determinant of its transpose: det(A^T) = det(A). This follows from the definition of the determinant via permutations, where transposing simply changes the sign conventions of the same set of terms in a way that cancels out. As a consequence, A is invertible if and only if A^T is invertible.
What is the transpose of a row vector or column vector?
Transposing a row vector (a 1 x n matrix) gives a column vector (an n x 1 matrix), and vice versa. This is the most common use of transposition in introductory linear algebra: converting between the two vector orientations. The dot product of two column vectors u and v can be written as u^T v, where u^T is the transpose of u.