Follow one set of numbers
The transpose of [[1,2,3],[4,5,6]] is [[1,4],[2,5],[3,6]].
Math calculator
Matrix Transpose Calculator
Exchange matrix rows and columns. A checkable formula accompanies transposed matrix instead of leaving an unexplained number.
Matrix Transpose inputs
Known quantities
Problems this can answer
Transposes appear in dot products, least squares, covariance, orthogonality, and coordinate changes.
How Matrix Transpose works
The transpose Aᵀ places entry aᵢⱼ at position aⱼᵢ and changes an m×n matrix into n×m.
The calculation treats matrix a according to their labels, not as interchangeable values in matrix transpose.
Reproducing the answer
Read each original column as one new row and check that transposing twice returns A.
Mistakes worth catching
Transposition is not inversion; it changes orientation without generally undoing a transformation. If the Matrix Transpose assumptions do not fit, consider x transpose A x.
Changing only one field helps distinguish a data-entry problem from the intended behavior of matrix transpose.
How the inputs shape the output
A field name is part of the formula. Match the problem's quantities to matrix a, then check that they share the scale assumed by the matrix transpose relationship.
How the result responds
A square matrix keeps its dimensions, while a rectangular matrix swaps them. This is a stronger check than judging the answer only by how many decimal places it shows.
Quadratic forms use a vector transpose to produce a scalar. The formula panel makes the chosen definition explicit.
Recording the answer
Match the reported precision to matrix a, not to the number of digits the browser can display. Preserve an exact form when it communicates the matrix transpose structure more clearly than a decimal.
The minimum audit trail is short: matrix a, their units, and the formula beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
Transposing twice
The Matrix Transpose setup records Matrix A. Apply the transpose again. The second transpose should recover the original entries and original dimensions exactly. Comparing the two corner positions also catches an incomplete row-column swap.
For Matrix Transpose, attach the coefficient convention and coordinate basis to the saved array. Transpose the reported matrix a second time; the original row and column arrangement should return exactly.
Questions about Matrix Transpose
What does Matrix Transpose calculate?
The transpose Aᵀ places entry aᵢⱼ at position aⱼᵢ and changes an m×n matrix into n×m.
When is Matrix Transpose useful?
Transposes appear in dot products, least squares, covariance, orthogonality, and coordinate changes.
What can make Matrix Transpose misleading?
Transposition is not inversion; it changes orientation without generally undoing a transformation.