1Swap rows and columns
The first row becomes the first column, the second row becomes the second column, and so on. The values themselves do not change.
A = [aᵢⱼ] ⇒ Aᵀ = [aⱼᵢ]
Free online matrix tool
Matrix Transpose Calculator
Transpose a matrix online in seconds. Swap rows and columns, work with square or rectangular matrices, and view the result alongside a clear explanation of how every entry moves.
Enter your matrix
Choose the number of rows and columns, then enter a number in each cell. The transpose will have the columns and rows reversed.
How to transpose a matrix
The transpose operation exchanges the row and column indices of every entry. It works for rectangular matrices as well as square matrices.
2Reverse the dimensions
If the original matrix has m rows and n columns, the transpose has n rows and m columns.
(m × n) → (n × m)
3Worked example
Start with this 2 × 3 matrix:
A = [[1, 2, 3], [4, 5, 6]]
The first row, [1, 2, 3], becomes the first column. The second row, [4, 5, 6], becomes the second column. The result therefore has three rows and two columns.
Aᵀ = [[1, 4], [2, 5], [3, 6]]
Click “Load example” in the calculator to try this matrix yourself.
Useful properties of the transpose
- Transpose twice: \((A^T)^T = A\). Applying the operation twice restores the original matrix.
- Transpose of a sum: \((A+B)^T = A^T+B^T\), when A and B have the same dimensions.
- Scalar multiplication: \((kA)^T = kA^T\), for a scalar k.
- Transpose of a product: \((AB)^T = B^T A^T\), whenever the product is defined.
- Symmetric matrices: a square matrix is symmetric when \(A^T=A\).
Frequently asked questions
What is the transpose of a matrix?
The transpose of a matrix is formed by exchanging its rows and columns. An entry originally in row i and column j moves to row j and column i. The transpose of A is written Aᵀ.
How do you calculate a matrix transpose?
Take each column of the original matrix and make it a row in the transposed matrix, keeping the entries in their original order. Equivalently, swap the row and column indices of every entry.
Can rectangular matrices be transposed?
Yes. The transpose is defined for rectangular as well as square matrices. For example, a 2 × 3 matrix becomes a 3 × 2 matrix.
Does transposing a matrix change its values?
No entries are added or removed. The same entries appear in different positions because the rows and columns are exchanged.
What is the transpose of the transpose?
Transposing a matrix twice returns the original matrix: (Aᵀ)ᵀ = A.
What is a symmetric matrix?
A square matrix is symmetric if its transpose equals itself, meaning Aᵀ = A. Its entries are mirrored across the main diagonal.
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