Matrix Transpose Calculator
The transpose of a matrix swaps its rows and columns, turning the entry in row i, column j into the entry in row j, column i. It is one of the most common operations in linear algebra, appearing in dot products, least squares regression and the definitions of symmetric and orthogonal matrices. This calculator accepts a matrix of any size, entered one row per line with entries separated by spaces or commas, and returns the transposed matrix along with the original and transposed dimensions. Ragged rows are detected and flagged.
Transpose definition
(A transpose) entry in row j, column i = A entry in row i, column j
An m by n matrix transposes to an n by m matrix
(A B) transpose = (B transpose)(A transpose)
Enter one row per line. Every row must contain the same number of numeric entries.
Transpose context
- Transposing swaps the row and column dimensions.
- A symmetric matrix equals its own transpose.
- The transpose of a product reverses the order of the factors.
- Row vectors and column vectors are transposes of each other.
- The transpose appears in the least squares normal equations A transpose A x equals A transpose b.
Matrix transpose: frequently asked questions
What is the transpose of a matrix?
The transpose of a matrix is formed by turning its rows into columns. The entry in row i, column j of the original becomes the entry in row j, column i of the transpose. A matrix with m rows and n columns transposes to one with n rows and m columns.
How do I enter the matrix?
Type one row per line. Separate the entries in a row with spaces or commas. Every row must have the same number of entries; if the rows are ragged, the calculator reports an error so you can fix the input.
What are the dimensions after transposing?
Transposing swaps the dimensions. If your matrix has m rows and n columns, the transpose has n rows and m columns. The calculator reports both the original and the transposed dimensions so you can confirm the swap.
Does transposing a symmetric matrix change it?
No. A symmetric matrix equals its own transpose, because the entry in row i, column j already equals the entry in row j, column i. Transposing it returns the same matrix unchanged.
What is the transpose used for?
The transpose appears throughout linear algebra: in dot products written as a row times a column, in the normal equations of least squares regression, in defining symmetric and orthogonal matrices, and in many identities such as the transpose of a product being the product of transposes in reverse order.
Official sources
- NIST Digital Library of Mathematical Functions: Elementary Algebra and Matrices.
- U.S. National Institute of Standards and Technology: NIST home, mathematical references.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.