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Linear Algebra

Matrix Transpose

The transpose of a matrix A, written Aᵀ, swaps its rows and columns: the entry in row i and column j of Aᵀ equals the entry in row j and column i of A. Enter your matrix and the calculator returns Aᵀ.

Matrix Transpose

Swap rows and columns to get Aᵀ.

Try:
Answer[[1, 4], [2, 5], [3, 6]]
  1. Matrix A2×3 [[1, 2, 3], [4, 5, 6]]
  2. RuleThe transpose Aᵀ swaps rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ.
  3. Transpose Aᵀ3×2 [[1, 4], [2, 5], [3, 6]]
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Worked examples

Transpose of [[1, 2, 3], [4, 5, 6]] Swap the rows and columns of a 2×3 matrix.

Frequently asked questions

Does the matrix have to be square?

No. Any m×n matrix has a transpose, which is an n×m matrix.

What is the transpose of a transpose?

(Aᵀ)ᵀ = A. Transposing twice returns the original matrix.

When is a matrix symmetric?

When A = Aᵀ. This is only possible for square matrices, and corresponds to the entries being mirrored across the main diagonal.

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