Matrix Transpose Calculator

Transpose a matrix and turn its rows into columns. Exact fractions, no rounding. Dimensions up to 6×6.

Each row of A becomes a column of Aᵀ A ᵀ Aᵀ A 2×3 → Aᵀ 3×2

How to Transpose a Matrix

Enter the entries of matrix A — whole numbers, decimals, and fractions like 1/2 are all accepted.

1

Set the dimensions

Pick the rows and columns for A, from 1×1 up to 6×6. Rows and columns can differ — transpose works on any rectangular matrix.

2

Fill in the entries

Type a value into each cell. The result updates as you type, and every value is kept as an exact fraction rather than a rounded decimal.

3

Read the result

The transposed matrix appears immediately — the first row of A becomes the first column of Aᵀ, and so on.

Frequently Asked Questions

What is the transpose of a matrix?
The transpose of a matrix flips it over its main diagonal, turning rows into columns and columns into rows. If A is an n-by-m matrix, its transpose is an m-by-n matrix in which the entry at row j, column i equals the entry at row i, column j of the original. So [[1, 2, 3], [4, 5, 6]] transposes to [[1, 4], [2, 5], [3, 6]]. Transposing twice returns the original matrix unchanged.
Does a matrix transpose change its dimensions?
The dimensions swap unless the matrix is square. An n-by-m matrix becomes an m-by-n matrix after transposing, so a 2x3 matrix becomes 3x2. Only when n equals m does the transpose keep the same shape as the original. The number of entries never changes, since n times m equals m times n.
What is the transpose used for?
Transposes appear throughout linear algebra. They are used to test whether a matrix is symmetric, which holds exactly when a matrix equals its own transpose, and to write the dot product of two column vectors as a matrix product. Least-squares regression is built on the normal equations, which multiply by a transpose on both sides, and transposes appear in QR and SVD decompositions and in machine-learning backpropagation formulas. The reversal rule is worth remembering: the transpose of AB is the transpose of B times the transpose of A.
Why does this calculator use fractions instead of decimals?
Matrix entries are stored as exact fractions using integer arithmetic, so a value like 1/3 stays exact instead of rounding to 0.3333. Transposing does not change any value, only its position, so no arithmetic error can creep in at this step. Keeping entries exact means the output matches whatever precision you typed in, with no rounding drift. It also means the transposed matrix can be pasted straight into the other matrix tools without losing accuracy.

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