Matrix Transpose Calculator

Background.

The Quanta matrix transpose calculator flips a matrix across its main diagonal: the first row becomes the first column, the second row becomes the second column, and so on. Type the matrix one row per line — entries separated by spaces or commas, up to 10 × 10 — and it returns the transpose along with its new shape, the number of entries, and a verdict on whether the original matrix is symmetric, skew-symmetric, both, or neither.

The transpose is the odd one out among the elementary matrix operations, and in a useful way. Adding, subtracting and scaling all leave the shape untouched: a 2 × 3 stays a 2 × 3. Transposing is the operation that changes it, turning an m × n matrix into an n × m one. That is exactly why it shows up wherever a shape has to be made to fit — a row vector becomes a column vector, a data matrix with observations in rows becomes one with observations in columns, and the product AᵀA turns any m × n matrix into a square, symmetric n × n one, which is the single most-used construction in least squares regression.

Four properties are worth committing to memory, and Encyclopedia of Mathematics lists all of them. Transposing a sum transposes each term: (A + B)ᵀ = Aᵀ + Bᵀ. Transposing a scalar multiple leaves the scalar alone: (αA)ᵀ = αAᵀ. Transposing twice gets you back where you started: (Aᵀ)ᵀ = A, which makes the transpose an involution. And the one people get wrong: transposing a product reverses the order, so (AB)ᵀ = BᵀAᵀ, not AᵀBᵀ. That reversal is not a quirk — it is forced by the shapes. If A is m × n and B is n × p, then AB is m × p and (AB)ᵀ is p × m; the only way to build a p × m matrix out of Aᵀ (n × m) and Bᵀ (p × n) is to put Bᵀ first. The same reversal applies to inverses, and for the same reason.

The symmetry verdict is the second thing this page gives you. A square matrix is symmetric when every pair of entries mirrored across the main diagonal is equal — Encyclopedia of Mathematics puts it as "a square matrix in which any two elements symmetrically positioned with respect to the main diagonal are equal to each other" — which is the same thing as saying A = Aᵀ. It is skew-symmetric when Aᵀ = −A, which forces every entry on the diagonal to be zero, since a number equal to its own negative must be zero. Symmetric matrices are the well-behaved ones: they have real eigenvalues, orthogonal eigenvectors, and a guaranteed diagonalisation, which is why covariance matrices, adjacency matrices of undirected graphs, stiffness matrices in structural analysis and kernel matrices in machine learning are all symmetric by construction. A non-square matrix is neither, and the calculator says so rather than guessing, because A and Aᵀ do not even have the same shape and so cannot be compared entry by entry.

One thing this page deliberately does not do is arithmetic. Every entry in the answer is an entry you typed, moved to a different position, so there is nothing to accumulate and nothing to lose. Entries are held as exact decimal values and normalised once to ten decimal places, and the symmetry check compares those same normalised values — the ones printed in the result — so the verdict can never contradict the matrix shown beside it.

What is matrix transpose calculator?

The transpose of a matrix A, written Aᵀ (sometimes A′ or Aᵗ), is the matrix obtained by interchanging its rows and columns: the entry in row i, column j of Aᵀ is the entry in row j, column i of A. Encyclopedia of Mathematics defines it as "the matrix obtained from a given (rectangular or square) matrix by interchanging the rows and the columns". If A is m × n then Aᵀ is n × m, so the transpose is the only elementary operation on a single matrix that changes its shape. Geometrically it is a reflection of the array across the main diagonal — the diagonal running from the top-left corner — and entries actually on that diagonal do not move. The transpose satisfies four algebraic rules that are used constantly: (A + B)ᵀ = Aᵀ + Bᵀ, (αA)ᵀ = αAᵀ for a scalar α, (AB)ᵀ = BᵀAᵀ with the order reversed, and (A⁻¹)ᵀ = (Aᵀ)⁻¹ when A is invertible. It is also an involution, meaning (Aᵀ)ᵀ = A. Two named classes are defined directly in terms of it. A square matrix is symmetric when A = Aᵀ, equivalently when aᵢⱼ = aⱼᵢ for every pair of indices. It is skew-symmetric (also called antisymmetric) when Aᵀ = −A; Encyclopedia of Mathematics defines this over a field of characteristic other than 2, and in that setting every diagonal entry must be zero, because aᵢᵢ = −aᵢᵢ forces 2aᵢᵢ = 0. The zero matrix is the only matrix that is both symmetric and skew-symmetric at once, since satisfying both conditions forces every entry to equal its own negative. In more abstract terms, the transpose is the matrix of the adjoint (or dual) of the linear map that A represents, which is the deeper reason the product rule reverses order: the dual of "do B then A" is "do the dual of A then the dual of B".

How to use this calculator.

1. 01Type the matrix into the box, one row per line, with entries separated by spaces or commas. "1 2 3" then "4 5 6" on the next line gives a 2 × 3 matrix. If you prefer a single line, semicolons also separate rows: "1,2,3;4,5,6".
2. 02Read the transpose in the hero panel. Semicolons separate rows, so [1 4; 2 5; 3 6] is a 3 × 2 matrix whose rows are (1, 4), (2, 5) and (3, 6).
3. 03Check the shape. The rows figure equals the number of columns you typed and the columns figure equals the number of rows — the shape flips. The entries count never changes, because transposing only rearranges numbers.
4. 04Read the symmetry verdict. It is only meaningful for a square matrix; for anything rectangular the calculator tells you the question does not apply rather than returning a misleading answer.
5. 05To confirm the involution property, copy the transpose back into the input box without the outer brackets and transpose it again — you get the original matrix back.
6. 06Decimals and negative numbers are fine. Entries are shown to at most ten decimal places, and the symmetry check compares those same displayed values, so two entries that print identically are treated as equal.

The formula.

The rule is a single swap of indices: whatever sits in row j, column i of A ends up in row i, column j of the transpose. Row 1 of A becomes column 1 of Aᵀ, row 2 becomes column 2, and so on, so an m × n matrix becomes an n × m one. Entries on the main diagonal — those with i = j — stay exactly where they are, which is why only square matrices can possibly equal their own transpose. Working through the example on this page: A = [1 2 3; 4 5 6] has rows (1, 2, 3) and (4, 5, 6), so the transpose has those as its columns, giving [1 4; 2 5; 3 6], a 3 × 2 matrix. Because no arithmetic is performed, the transpose is exact in every case — the calculator holds each entry as an arbitrary-precision decimal, normalises it once to ten decimal places, and moves it. The four algebraic properties follow from the index swap. For a sum, the (i, j) entry of (A + B)ᵀ is the (j, i) entry of A + B, which is aⱼᵢ + bⱼᵢ, which is the (i, j) entry of Aᵀ + Bᵀ. For a scalar multiple the same argument gives (αA)ᵀ = αAᵀ. Transposing twice returns the original, because swapping the indices twice is the identity. The product rule is the interesting one: (AB)ᵀ = BᵀAᵀ, with the order reversed. The (i, j) entry of (AB)ᵀ is the (j, i) entry of AB, which is the sum over k of aⱼₖbₖᵢ. Reading that same sum as a row-by-column product of Bᵀ and Aᵀ gives the (i, j) entry of BᵀAᵀ. The shapes confirm it: if A is m × n and B is n × p, then (AB)ᵀ is p × m, and the only conformable way to build that from Bᵀ (p × n) and Aᵀ (n × m) is Bᵀ first. The symmetry verdict is a direct comparison rather than a computed threshold. The calculator checks aᵢⱼ against aⱼᵢ for every pair, and separately against −aⱼᵢ, on the same ten-decimal-place values it prints. If all pairs match, the matrix is symmetric; if all pairs match under negation, it is skew-symmetric; if both hold, every entry is zero and the matrix is the zero matrix, which is the only one that qualifies as both; otherwise it is neither. For a non-square matrix the comparison is not defined at all, and the calculator reports that rather than inventing a verdict.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Encyclopedia of Mathematics (EMS Press / Springer), article "Transposed matrix" — defines the transpose as the matrix obtained by interchanging rows and columns, and lists the four properties used on this page, including (AB)ᵀ = BᵀAᵀ. Retrieved 2026-07-29; open access.
2. [2]Encyclopedia of Mathematics (EMS Press / Springer), article "Symmetric matrix" — "a square matrix in which any two elements symmetrically positioned with respect to the main diagonal are equal to each other". Retrieved 2026-07-29; open access.
3. [3]Encyclopedia of Mathematics (EMS Press / Springer), article "Skew-symmetric matrix" — "A square matrix A over a field of characteristic ≠ 2 such that Aᵀ = −A". The zero diagonal follows from this rather than being a separate requirement. Retrieved 2026-07-29; open access.
4. [4]NIST Digital Library of Mathematical Functions, §1.3(iv) "Matrices as Linear Operators" — the operator framing that explains why the transpose reverses the order of a product. Retrieved 2026-07-29; open access.
5. [5]MIT OpenCourseWare 18.06 Linear Algebra, Spring 2010, Prof. Gilbert Strang — lectures on transposes, symmetric matrices and the AᵀA construction behind least squares. CC BY-NC-SA 4.0; retrieved 2026-07-29; open access.