Matrix Transpose Calculator
Free matrix transpose calculator. Swap the rows and columns of any matrix to get A transpose, with the result shown instantly for 2x2, 2x3, 3x3, 3x4 and larger matrices.
Turn rows into columns instantly. Enter any matrix — square or rectangular — and the transpose appears alongside it, with a note on where each entry moved.
How to transpose a matrix
- 1
Set the dimensions
Enter the number of rows and columns. Unlike determinants, the transpose works on any shape.
- 2
Enter the matrix
Fill in the grid. Fractions and negative numbers are both accepted.
- 3
Read Aᵀ
The transpose appears beside the original, with each row of Aᵀ built from the matching column of A.
Transpose rules
- (Aᵀ)ᵀ = A — transposing twice returns the original.
- (A + B)ᵀ = Aᵀ + Bᵀ — transposition distributes over addition.
- (kA)ᵀ = kAᵀ — scalars pass straight through.
- (AB)ᵀ = BᵀAᵀ — products reverse order.
- det(Aᵀ) = det(A) — the determinant is unchanged.
- AᵀA is always symmetric — the basis of least squares and the normal equations.
Transpose calculators by size
Where transposes are used
Least squares and regression. The normal equations AᵀAx = Aᵀb are the standard way to fit a line through data that does not fit exactly.
Dot products. For column vectors, uᵀv is exactly the dot product, which makes transposition the bridge between matrix notation and geometry.
Covariance matrices. Any matrix of the form AᵀA is symmetric and positive semi-definite, which is what makes covariance matrices well behaved.
Graph theory. Transposing an adjacency matrix reverses every edge in a directed graph.
Frequently asked questions
What is the transpose of a matrix? ▾
The transpose Aᵀ is the matrix you get by reflecting A across its main diagonal: the first row becomes the first column, the second row becomes the second column, and so on. The entry at position (i, j) moves to position (j, i).
How do the dimensions change when you transpose? ▾
They swap. An m x n matrix transposes to an n x m matrix, so a 2x3 becomes a 3x2. Only square matrices keep the same shape.
What is a symmetric matrix? ▾
A square matrix that equals its own transpose, so A = Aᵀ. Symmetric matrices have real eigenvalues and can always be diagonalized by an orthogonal matrix, which is why they turn up everywhere in statistics and physics.
Is (AB)ᵀ the same as AᵀBᵀ? ▾
No — the order reverses: (AB)ᵀ = BᵀAᵀ. This trips people up constantly. Transposing a product flips the multiplication order, the same way inverting a product does.
What is the transpose of a transpose? ▾
You get the original matrix back: (Aᵀ)ᵀ = A. Transposition is its own inverse.
About this Calculator
Free matrix transpose calculator. Swap the rows and columns of any matrix to get A transpose, with the result shown instantly for 2x2, 2x3, 3x3, 3x4 and larger matrices.