---
title: "Matrix Transpose Calculator — with Symmetry Check"
description: "Transpose any matrix (2×2 to 5×5) — swap rows and columns with full element-by-element steps. Detects symmetric matrices. Covers A^T notation, (AB)^T = B^T A^T, and applications in ML, physics, and statistics."
url: "https://worldcalculators.org/calculators/matrix-transpose/"
canonical: "https://worldcalculators.org/calculators/matrix-transpose/"
---

# Matrix Transpose Calculator

Transpose any matrix (2×2 to 5×5) — swap rows and columns with full element-by-element steps. Detects symmetric matrices. Covers A^T notation, (AB)^T = B^T A^T, and applications in ML, physics, and statistics.

## Quick answer

Matrix Dimensions

## What Is the Transpose?

The transpose of a matrix A is formed by writing its rows as columns. If A has shape m×n, then A^T has shape n×m. This operation is fundamental to linear algebra and appears in every branch of science and engineering.
A square matrix where A = A^T is called symmetric . Symmetric matrices have real eigenvalues and orthogonal eigenvectors (Spectral Theorem). They appear everywhere:

## Frequently Asked Questions

The transpose of A, written A^T, swaps rows and columns: A^T[i][j] = A[j][i]. A 3×4 matrix transposes to a 4×3 matrix.
A square matrix where A = A^T. This means A[i][j] = A[j][i] for all positions. Covariance matrices and Gram matrices (X^T X) are always symmetric.

Source: https://worldcalculators.org/calculators/matrix-transpose/
Markdown mirror of the HTML page. Prefer this URL for RAG; the interactive calculator still lives on the HTML page.