Transpose Matrix Calculator

About Transpose Matrix Calculator

Transposing a matrix flips it along its main diagonal, turning rows into columns and columns into rows. The entry at row 2, column 5 moves to row 5, column 2. An m x n matrix becomes n x m, so unlike the determinant and inverse, transposition works on any matrix and not just square ones. It is the cheapest matrix operation there is, requiring no arithmetic at all, only rearrangement. It appears constantly as a supporting step rather than an end in itself. In statistics, multiplying a data matrix by its own transpose produces the covariance structure. In graphics, the transpose of a rotation matrix is its inverse, which makes undoing a rotation free. Transposing twice returns the original matrix, and the transpose of a product reverses the order, so the transpose of AB is the transpose of B times the transpose of A. A matrix equal to its own transpose is called symmetric, which is a property that unlocks faster algorithms in numerical work. Enter any matrix, of any shape, and the result appears immediately. For the other operations see Matrix Calculator.

How to use Transpose Matrix Calculator

  1. Select an operation from the tab bar (Add, Multiply, etc.)
  2. Set matrix dimensions and enter values in the grid cells
  3. See the result matrix or scalar instantly

Frequently Asked Questions

What does transposing a matrix do?
It flips the matrix along its main diagonal, so rows become columns and columns become rows. The entry at row i, column j moves to row j, column i.
Can you transpose a non-square matrix?
Yes. Transposition works on any shape. An m x n matrix becomes n x m, so a 2x5 matrix transposes to a 5x2 one. Only the determinant and inverse require a square matrix.
What happens if you transpose twice?
You get back the original matrix. Transposition is its own inverse operation, since flipping along the diagonal twice returns every entry to where it started.
What is a symmetric matrix?
One that equals its own transpose, meaning the entry at row i, column j always matches the entry at row j, column i. Symmetric matrices have useful properties that let numerical algorithms run faster.
What is the transpose of a product?
It reverses the order: the transpose of AB equals the transpose of B multiplied by the transpose of A. Getting this order wrong is a common slip in derivations.

Related Tools

Also Available As