↔ Transpose Calculator

Matrix Transpose Calculator

Transpose any m×n matrix (swap rows and columns). 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 SymPy Engine…
Matrix A
Quick Examples

Matrix Transpose

The transpose of a matrix A is written Aᵀ and is formed by swapping its rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ

If A is m×n, then Aᵀ is n×m.

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

Key Properties

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