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 (written A with a superscript T) is an m-by-n matrix where entry [j,i] of the transpose equals entry [i,j] of the original.
Does a matrix transpose change its dimensions?
Yes, unless the matrix is square. An n-by-m matrix becomes an m-by-n matrix after transposing — a 2x3 matrix becomes 3x2. Only for square matrices (n equals m) does the transpose keep the same dimensions as the original.
What is the transpose used for?
Transposes are used throughout linear algebra: to check whether a matrix is symmetric (A equals its own transpose), to compute the dot product of vectors as matrix multiplication, in least-squares regression, and in many matrix decompositions and machine learning formulas.
Why does this calculator use fractions instead of decimals?
Matrix entries are stored as exact fractions using integer arithmetic, so results like 1/3 stay exact instead of rounding to 0.333... Transposing does not change values, only positions, but keeping entries exact means the output matches whatever precision you typed in without any rounding drift.

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