Math
Matrix Transpose Calculator
Enter a matrix to flip it across its main diagonal.
The calculator shows Aᵀ with each row turned into a column, tells you whether a square matrix is symmetric (Aᵀ = A) or skew-symmetric (Aᵀ = -A), computes AᵀA and AAᵀ, and lists the transpose identities you need for proofs and homework.
One row per line. Separate entries with spaces or commas. Integers, decimals, and fractions like 1/3 all work. Up to 8 x 8.
Try an example
Result
Transpose
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- Size of Aᵀ
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- Symmetric?
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- Classification
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Study path
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What the transpose is
The The matrix you get by flipping rows into columns and columns into rows. of an m x n A rectangular grid of numbers arranged in rows and columns. A is the n x m matrix Aᵀ whose rows are the columns of A. Entry (i, j) of Aᵀ is entry (j, i) of A. Transposing twice returns the original matrix.
Transpose rules
| Rule | Formula |
|---|---|
| Double transpose | (Aᵀ)ᵀ = A |
| Sum | (A + B)ᵀ = Aᵀ + Bᵀ |
| Scalar | (kA)ᵀ = kAᵀ |
| Product (order reverses) | (AB)ᵀ = BᵀAᵀ |
| Inverse | (A⁻¹)ᵀ = (Aᵀ)⁻¹ |
| Determinant | det(Aᵀ) = det(A) |
| Trace | trace(Aᵀ) = trace(A) |
Symmetric and related matrices
- Symmetric: Aᵀ = A. Entries mirror across the diagonal. Real symmetric matrices have real A number saying how much a matrix stretches vectors along one special direction. and orthogonal eigenvectors.
- Skew-symmetric: Aᵀ = -A. The diagonal must be zero.
- Orthogonal: AᵀA = I, so Aᵀ = A⁻¹. Rotations and reflections are orthogonal.
- For any A, both AᵀA and AAᵀ are symmetric; they appear in least squares and in the singular value decomposition.
How to use it
- Enter the matrix, one row per line. It does not need to be square.
- Read Aᵀ in the panel and check the size: an m x n input gives an n x m output.
- Use the symmetric and skew-symmetric flags and the AᵀA, AAᵀ products for the follow-up questions in your assignment.
Common mistakes
- Writing (AB)ᵀ = AᵀBᵀ. The order reverses: (AB)ᵀ = BᵀAᵀ.
- Calling a rectangular matrix symmetric. Only square matrices can equal their transpose.
- Confusing transpose with inverse. They agree only for orthogonal matrices.
How to type a matrix into this calculator
- Put each row on its own line. Separate the entries in a row with spaces or commas: 2 1 1 or 2, 1, 1.
- Fractions such as 1/3 and decimals such as 0.25 are accepted and kept exact, so 1/3 stays 1/3 instead of becoming 0.3333.
- Every row must have the same number of entries. A row with a missing entry produces a clear error instead of a wrong answer.
- The largest size supported is 8 x 8. Brackets and semicolons are ignored, so [1 2; 3 4] works too.
Worked examples
Rectangular 2 x 3
[1 2 3; 4 5 6]ᵀ is 3 x 2
Transpose
[1, 4; 2, 5; 3, 6]
Symmetric matrix
[2 1; 1 2] equals its transpose
Transpose
[2, 1; 1, 2]
Skew-symmetric matrix
[0 -3; 3 0]ᵀ = -[0 -3; 3 0]
Transpose
[0, 3; -3, 0]
Column vector becomes a row vector
(1, 2, 3)ᵀ
Transpose
[1, 2, 3]
Fractions
[1/2 -1/3; 0 5]ᵀ
Transpose
[1/2, 0; -1/3, 5]
Frequently asked questions
What does transposing a matrix do?+
It swaps rows and columns. The first row becomes the first column, the second row becomes the second column, and so on. An m x n matrix becomes n x m.
How do I know if a matrix is symmetric?+
Compare it with its transpose. If every entry a_ij equals a_ji, the matrix is symmetric. The calculator checks this automatically for square inputs.
What is (AB) transpose?+
(AB)ᵀ = BᵀAᵀ. The order of the factors reverses, which is easy to forget.
Is the transpose the same as the inverse?+
Only for orthogonal matrices, where AᵀA = I. In general the transpose and inverse are different matrices.
Why is AᵀA always symmetric?+
Because (AᵀA)ᵀ = Aᵀ(Aᵀ)ᵀ = AᵀA. The same argument shows AAᵀ is symmetric. These products show up in least-squares fitting.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026