Matrix Transpose Calculator (Aᵀ)

Matrix transpose calculator that computes Aᵀ for any m×n matrix. Supports custom size, random fill, copying result back, and LaTeX output with step-by-step explanation of the transpose operation.

Full original guide (expanded)

Matrix Transpose Calculator (Aᵀ)

Build an m×n matrix, compute its transpose Aᵀ, and get a clean numeric table plus a ready-to-copy LaTeX representation. Ideal for linear algebra homework, coding, and documentation.

Supports custom size, rectangular matrices, random fill, identity/zeros presets, and copying the result back into the input grid.

Enter a matrix and compute its transpose

Maximum size: 10×10. Use the buttons below to quickly fill common patterns (identity, zeros, random).

Input matrix A (m×n)

What is the transpose of a matrix?

Given an m×n matrix \(A\), its transpose, denoted by \(A^T\) (or sometimes \(A'\)), is the n×m matrix obtained by swapping rows and columns. Formally:

\[ (A^T)_{ij} = A_{ji} \]

In other words, the entry in the i-th row and j-th column of the transpose comes from the j-th row and i-th column of the original matrix.

Example

Start from the 2×3 matrix:

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \]

The transpose is a 3×2 matrix:

\[ A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} \]

Every row of \(A\) becomes a column of \(A^T\), and every column of \(A\) becomes a row of \(A^T\).

Key properties of the transpose

  • \((A^T)^T = A\)
  • \((A + B)^T = A^T + B^T\)
  • \((cA)^T = cA^T\) for any scalar \(c\)
  • \((AB)^T = B^T A^T\) whenever the product \(AB\) is defined

These identities are fundamental in linear algebra, especially when working with inner products, adjoint operators, and least squares problems.

Why is the transpose useful?

  • Symmetric matrices: matrices for which \(A^T = A\), central in optimization and statistics.
  • Orthogonal matrices: where \(A^{-1} = A^T\), modeling rotations and rigid transformations.
  • Data manipulation: switching between “samples as rows” and “features as rows”.
  • Programming: many APIs and algorithms expect a particular orientation of vectors and matrices; transpose adapts data accordingly.

FAQ – Matrix transpose

What happens to the determinant when I transpose a matrix?
For a square matrix, the determinant of the transpose equals the determinant of the original matrix: \(\det(A^T) = \det(A)\). This is consistent with the idea that the transpose does not change the eigenvalues, only the orientation of the basis.
Does transposing change the eigenvalues of a matrix?
No. A matrix and its transpose have the same characteristic polynomial, so they share the same eigenvalues (with the same algebraic multiplicities).
How is transpose implemented in numerical libraries?
In low-level libraries, the transpose is often implemented either by changing the way indices are interpreted (strides) or by explicitly copying data into a new array with swapped axes. High-level libraries like NumPy, MATLAB, and Wolfram Language expose convenient operators such as A' or A.T that internally rely on these mechanisms.
Audit: Complete
Formula (LaTeX) + variables + units
This section shows the formulas used by the calculator engine, plus variable definitions and units.
Formula (extracted LaTeX)
\[(A^T)_{ij} = A_{ji}\]
(A^T)_{ij} = A_{ji}
Formula (extracted LaTeX)
\[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\]
A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}
Formula (extracted LaTeX)
\[A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}\]
A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}
Formula (extracted LaTeX)
\[','\\]
','\
Formula (extracted text)
\[ (A^T)_{ij} = A_{ji} \]
Formula (extracted text)
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \]
Formula (extracted text)
\[ A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} \]
Variables and units
  • T = property tax (annual or monthly depending on input) (currency)
Sources (authoritative):
Changelog
Version: 0.1.0-draft
Last code update: 2026-01-19
0.1.0-draft · 2026-01-19
  • Initial audit spec draft generated from HTML extraction (review required).
  • Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
  • Confirm sources are authoritative and relevant to the calculator methodology.
Verified by Ugo Candido on 2026-01-19
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Formulas

(Formulas preserved from original page content, if present.)

Version 0.1.0-draft
Citations

Add authoritative sources relevant to this calculator (standards bodies, manuals, official docs).

Changelog
  • 0.1.0-draft — 2026-01-19: Initial draft (review required).