CALCULATE WITH CONFIDENCE

Linear Algebra & Matrices

Matrix Transpose Calculator

Exchange rows and columns to form the exact transpose of a matrix.

Linear Algebra & Matrices

Enter matrix values

Exact rational arithmetic
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example
Matrix A

Use ordinary signed decimals. Fractions, commas, and scientific notation are not accepted in this first batch.

Exact result

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Understand the subject

What is a transpose?

The transpose exchanges rows and columns. An entry at row i, column j moves to row j, column i.

Use it to reverse a matrix’s row/column viewpoint and build symmetric products.

The relationship

The defining relationship

See the structure

What the calculation is doing

Worked interpretation

Read the result in context

(Aᵀ)ᵢⱼ = aⱼᵢ

Interpret with care

Important boundary

Transpose changes the shape of a rectangular matrix.

Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.

Quick guide

How to use this calculator

  1. Choose A's dimensions.
  2. Enter every matrix value.
  3. Read Aᵀ with its row and column counts exchanged.

Calculation method

Reflect entries across the main diagonal

The entry aᵢⱼ becomes the entry in row j and column i of Aᵀ.

Worked example

Transpose a rectangular matrix

The two rows of [[1,2,3],[4,5,6]] become three columns.

Aᵀ = [[1,4],[2,5],[3,6]]

Supported inputs

Precision and limits

Matrix size

Each input matrix may contain 1–6 rows and 1–6 columns.

Entry precision

Each signed decimal may contain up to 30 digits and 15 decimal places.

Exactness

Finite decimals are converted to reduced fractions before calculation; matrix arithmetic is not rounded.