Matrix Transpose Calculator — Aᵀ

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Walk-through

How to Use This Calculator

3 steps

1

Set the matrix dimensions

Choose the number of rows and columns for matrix A using the Rows and Columns dropdowns. Any shape from 1×1 up to 5×5 is supported, including non-square (rectangular) matrices and single-row or single-column vectors.

2

Enter the matrix entries

Fill in each cell of matrix A with a number. The transpose Aᵀ recalculates instantly in the result panel as you type — no need to press a calculate button.

3

Read the transpose and symmetric check

The result card shows Aᵀ with rows and columns swapped from A. Switch to the Symmetric Check tab to see whether A equals its own transpose (only possible for square matrices), or the Dimensions tab to confirm how the shape changed.

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Reference

Formula & Methodology

2 formulas

Transpose

Aᵀ[j][i] = A[i][j]

The transpose Aᵀ is formed by swapping rows and columns: the entry in row i, column j of A becomes the entry in row j, column i of Aᵀ. An m×n matrix (m rows, n columns) becomes an n×m matrix after transposing.

Symmetric matrix test

A = Aᵀ

A square matrix A is symmetric when every entry mirrors across the main diagonal, i.e. A[i][j] = A[j][i] for all i, j. Symmetry is only defined for square matrices — a rectangular matrix can never equal its own transpose because the two have different dimensions.

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Glossary

Key Terms Explained

6 terms

Transpose (Aᵀ) The matrix formed by flipping A over its main diagonal, so its rows become columns and its columns become rows. Written as Aᵀ or A′. An m×n matrix transposes to an n×m matrix.

Symmetric matrix A square matrix that is equal to its own transpose (A = Aᵀ). Every off-diagonal entry has a matching mirror value on the opposite side of the main diagonal, e.g. A[1][2] = A[2][1].

Rows The horizontal lines of entries in a matrix, read left to right. A matrix with m rows has m horizontal entry groups; the row count becomes the column count of the transpose.

Columns The vertical lines of entries in a matrix, read top to bottom. A matrix with n columns has n vertical entry groups; the column count becomes the row count of the transpose.

Matrix dimensions The size of a matrix expressed as rows × columns (m×n). Transposing swaps the two numbers, so a 2×3 matrix becomes a 3×2 matrix while its total entry count (6) stays the same.

Row/column vector A matrix with only one row (1×n, a row vector) or only one column (m×1, a column vector). Transposing a row vector produces a column vector with the same entries, and vice versa.

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Scenarios

Real-World Examples

3 worked examples

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Rectangular matrix

2×3 matrix

Matrix A [[1,2,3],[4,5,6]] Dimensions 2 rows × 3 columns

A is 2×3, so Aᵀ becomes 3×2: [[1,4],[2,5],[3,6]]. Row 1 of A (1, 2, 3) becomes column 1 of Aᵀ. The entry count stays the same (6), only the shape changes — and because rows ≠ columns, a symmetric check isn't defined for this matrix.

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Symmetric matrix

3×3 symmetric example

Matrix A [[2,-1,0],[-1,3,4],[0,4,5]] Dimensions 3 rows × 3 columns

Every off-diagonal pair mirrors: A[1][2]=-1 matches A[2][1]=-1, and A[2][3]=4 matches A[3][2]=4. Because A equals its own transpose, the Symmetric Check tab reports A = Aᵀ — a common shape for covariance matrices and undirected-graph adjacency matrices.

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Vector transpose

1×3 row vector

Matrix A [[7,8,9]] Dimensions 1 row × 3 columns

A single row vector [7, 8, 9] transposes into a 3×1 column vector with the same three entries stacked vertically. This row-to-column flip is exactly what happens when converting a data point into the column-vector form required by many linear algebra formulas.

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Deep Dive

Understanding the Matrix Transpose Calculator

3 chapters · 4 min read

The transpose of a matrix flips it over its main diagonal, turning rows into columns and columns into rows — one of the most frequently used operations in linear algebra. This calculator computes Aᵀ for any matrix up to 5×5, checks whether a square matrix is symmetric, and shows exactly how the dimensions change.

How the Matrix Transpose Calculator works

Transposing a matrix follows a single rule: Aᵀ[j][i] = A[i][j]. Every entry in row i, column j of the original matrix A moves to row j, column i of the result. Practically, this means you can read down each column of A to get the rows of Aᵀ. The calculator applies this swap to every cell you enter, updating live as you type, and works for any rectangular shape from 1×1 up to 5×5.

Inputs and what they mean

Set the Rows and Columns dropdowns first — they define the shape of matrix A, from a single-cell 1×1 matrix up to a full 5×5 grid. Rows and columns don't need to match: rectangular shapes like 2×3 or 4×1 are fully supported alongside square matrices. Once the grid appears, enter any real number (positive, negative, or decimal) into each cell. The Presets strip offers a quick rectangular example, a symmetric 3×3 example, and a single-row vector to explore without typing from scratch.

Limits and edge cases

The Symmetric Check tab only applies to square matrices (rows = columns) — for a rectangular matrix, A and Aᵀ have different dimensions and can never be equal, so the calculator reports symmetry as not applicable rather than false. Applying the transpose twice always returns the original matrix, (Aᵀ)ᵀ = A, which is a useful way to sanity-check your entries. This tool caps at 5×5 for readability; larger matrices are typically handled with dedicated linear algebra software rather than a manual grid.

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Questions

Frequently Asked Questions

6 questions

What is the formula for a matrix transpose?+

The transpose swaps rows and columns using Aᵀ[j][i] = A[i][j]: the entry at row i, column j of the original matrix moves to row j, column i of the result. In practice, the first row of A becomes the first column of Aᵀ, the second row becomes the second column, and so on. An m×n matrix (m rows, n columns) always becomes an n×m matrix after transposing.

What makes a matrix symmetric?+

A matrix is symmetric when it equals its own transpose, A = Aᵀ. That requires the matrix to be square (same number of rows and columns) and for every entry to mirror across the main diagonal — for example, the entry in row 1, column 2 must equal the entry in row 2, column 1. Symmetric matrices show up frequently in covariance matrices, distance matrices, and adjacency matrices for undirected graphs.

How do the dimensions change when you transpose a matrix?+

Transposing swaps the row and column counts: an m×n matrix becomes an n×m matrix. For example, a 2×4 matrix (2 rows, 4 columns) transposes into a 4×2 matrix (4 rows, 2 columns). The total number of entries never changes — only the arrangement of rows versus columns does. Square matrices (where rows = columns) keep the same dimensions after transposing, even though the entries themselves still move.

What is (Aᵀ)ᵀ — the transpose of a transpose?+

Transposing a matrix twice always returns the original matrix: (Aᵀ)ᵀ = A. The first transpose swaps rows and columns; the second transpose swaps them back to their original positions. This property is a quick way to double-check a manual transpose calculation — if transposing your result doesn't recreate the original matrix, an entry was likely misplaced.

How do you transpose a row or column vector?+

A row vector (a matrix with a single row, shape 1×n) transposes into a column vector (shape n×1) with the same entries stacked vertically instead of horizontally, and the reverse is true for a column vector. This row-to-column conversion is a common step when a formula requires vectors in a specific orientation, such as computing a dot product or setting up a system of linear equations.

What is the matrix transpose used for?+

The transpose appears throughout linear algebra: it's required to compute a dot product between two vectors written as matrices, to test whether a matrix is symmetric or orthogonal, to derive the normal equations in least-squares regression, and to convert between row-vector and column-vector conventions when combining formulas from different sources. It's one of the most common matrix operations alongside addition, multiplication, and inversion.

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Transpose (Aᵀ)