↔ Transpose Calculator

Matrix Transpose Calculator

The transpose of a matrix is obtained by swapping its rows and columns, and this calculator computes it for any m×n matrix with a symmetry check included. Shows step-by-step element mapping, symmetry check, and verifies (Aᵀ)ᵀ = A.

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Matrix Transpose

Aᵀ — swaps rows and columns

✓ Any Size✓ Symmetry Check✓ Free⚡ Loading precision engine…
Matrix A
Quick Examples

Matrix Transpose

The transpose of a matrix, written Aᵀ, is what you get by flipping the matrix over its diagonal — every row becomes a column, and every column becomes a row. Unlike most matrix operations, transpose works on matrices of any shape, not just square ones.

$$\begin{pmatrix}a & b & c \\ d & e & f\end{pmatrix}^T = \begin{pmatrix}a & d \\ b & e \\ c & f\end{pmatrix}$$

Key Properties

  • (Aᵀ)ᵀ = A
  • (A+B)ᵀ = Aᵀ + Bᵀ
  • (AB)ᵀ = BᵀAᵀ
  • det(Aᵀ) = det(A)
  • If A = Aᵀ, A is called symmetric

Worked Example (2×3)

For A = [[1,2,3],[4,5,6]] (2 rows, 3 columns), the transpose Aᵀ = [[1,4],[2,5],[3,6]] (3 rows, 2 columns). Notice the first row of A (1,2,3) became the first column of Aᵀ.

Applications

  • Converting between row and column vectors
  • Required step in many formulas, such as finding a matrix's inverse or checking if it's symmetric
  • Used constantly in statistics and machine learning, such as calculating ATA for least-squares regression

Frequently Asked Questions

No — transpose never changes any of the numbers, it only changes their position, moving each value from row i, column j to row j, column i.
A symmetric matrix is one that stays exactly the same after being transposed — meaning A = Aᵀ. This can only happen with square matrices.
Yes — transpose is one of the few matrix operations that works on any size matrix, including rectangular ones. A 2×3 matrix becomes a 3×2 matrix after transposing.

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