Last updated: August 1, 2026
Matrix Transpose Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
Enter a matrix in row format to swap its rows and columns and compute the transpose. The calculator returns the transposed matrix plus the original and transposed dimensions, which is useful when checking symmetry, preparing products, or converting between row and column viewpoints.
Enter any matrix in row form to swap its rows and columns and compute the transpose.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
The transpose swaps row and column positions, so (Aᵀ)ij = aji.
Where:
- A=original matrix
- Aᵀ=transpose of the matrix
Worked Examples
Transpose a 2×3 matrix
Swap rows and columns.
- 1Take the first column as the first row of the transpose.
- 2Repeat for each remaining column.
Transpose a square matrix
A 2×2 matrix stays square after transposition.
- 1Swap the off-diagonal entries.
- 2Keep the diagonal entries fixed.
Transpose a row vector
A 1×3 row becomes a 3×1 column.
- 1Treat the input as one row.
- 2Write each entry as its own row.
Transpose twice conceptually
Applying transpose again restores the original matrix.
- 1Compute the transpose once.
- 2Observe that another transpose would undo it.
Introduction
Swap rows and columns to form the transpose of a matrix and confirm the new dimensions.
Overview
Swap rows and columns to form the transpose of a matrix and confirm the new dimensions.
Transposed matrix
Symmetry checks
Preparing matrix products
Least-squares setup
Input format
Enter any rectangular matrix in row form.
Separate rows with semicolons
Separate entries with commas or spaces
Brackets are optional
Keep row lengths consistent
Formula and method
The transpose swaps row and column positions, so (Aᵀ)ij = aji.
- A matters:
original matrix.
- Aᵀ matters:
transpose of the matrix.
Validation rules
The calculator uses deterministic validation before it computes the final result.
The matrix must be rectangular and nonempty
Every entry must be finite
Rows must have equal length
Transposition is allowed for square and rectangular matrices
Interpreting the outputs
Use the main result first, then the secondary fields to verify the structure and meaning of the answer.
Rows become columns
m×n becomes n×m
Diagonal entries stay in place
A transpose applied twice returns the original matrix
Common use cases
These are typical reasons to use this calculator in study or applied work.
Symmetry checks
Preparing matrix products
Least-squares setup
Switching between row and column views
FAQs
Can I transpose a rectangular matrix?
Yes. Any rectangular matrix can be transposed.
What happens to the dimensions?
An m×n matrix becomes an n×m matrix.
Do diagonal entries change?
No. Entries on the main diagonal stay in the same positions.
What is Aᵀᵀ?
Transposing twice gives back the original matrix.
Can I use a single row or single column?
Yes. Row vectors become columns and column vectors become rows.
Can I use decimals?
Yes. Any finite real entries are supported.