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Matrix Transpose Calculator

Work out matrix transpose instantly with clear inputs, formula shown and shareable results.

About Matrix Transpose Calculator

Transposing reflects a matrix across its main diagonal, turning rows into columns. An m × n matrix becomes n × m, and the trace of a square matrix is unchanged.

Formulas

Transpose

(Aᵀ)ᵢⱼ = Aⱼᵢ

Product rule

(AB)ᵀ = BᵀAᵀ

Worked example

With the default values below, Matrix Transpose Calculator returns Aᵀ of [1, 4] [2, 5] [3, 6]. Change any input above and every figure here is recalculated in your browser.

Inputs used

Matrix, row by row
1, 2, 3, 4, 5, 6
Columns per rowaccepts 1–8
3

Results

Aᵀ
[1, 4] [2, 5] [3, 6]
Shape of the transpose
3 by 2
Symmetric
No, A is not symmetric
Trace (0 if not square)
0.000000

Step by step

  1. 1Values used
  2. 2Transpose
  3. 3Product rule
  4. 4Aᵀ= [1, 4] [2, 5] [3, 6]
  5. 5Shape of the transpose= 3 by 2
  6. 6Symmetric= No, A is not symmetric
  7. 7Trace (0 if not square)= 0.000000

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Transpose [[1,2,3],[4,5,6]].

[[1, 4], [2, 5], [3, 6]] — a 3 × 2 matrix.

Why does (AB)ᵀ reverse the order?

Because the row-column pairing swaps, so the last factor must be applied first for the dimensions to line up.

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