How this calculator works
The entry in row i and column j of the original matrix becomes the entry in row j and column i of the transpose. A matrix with m rows and n columns becomes a matrix with n rows and m columns. The transpose of a transpose returns the original matrix. A square matrix that equals its own transpose is called symmetric.
Formula
(A^T)ij = Aji. The transpose swaps the row and column indices of every entry.
Worked example
Transposing the 2 by 3 matrix with rows 1, 2, 3 and 4, 5, 6 gives the 3 by 2 matrix with rows 1, 4; 2, 5; 3, 6. The symmetric matrix with rows 2, 1 and 1, 3 is unchanged by the transpose.
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Frequently asked questions
Does the transpose change the determinant?
No. For a square matrix, det(A transpose) equals det(A), so transposing never changes invertibility.
What happens to a column vector?
A column vector becomes a row vector, and a row vector becomes a column vector. Set one dimension to 1 to use this tool that way.
Which entries stay in place?
Entries on the main diagonal keep their position. Every off-diagonal entry swaps with its mirror image across the diagonal.