Calculators / Linear Algebra / Transpose of a Matrix Calculator

Transpose of a Matrix Calculator

Published on: October 12 2025

This Transpose of a Matrix Calculator helps you find the transpose of a matrix and shows each step clearly. It works by swapping the rows and columns so that each entry in position (i, j) moves to position (j, i). This makes it useful for checking answers, understanding how the transpose of a matrix works, and practising linear algebra step by step.

Step-by-step method

  1. Write the matrix A.
  2. Swap rows and columns to form Aᵀ.
  3. Read off the transposed matrix.

Transpose formula

( AT )ij = aji,
where A =
a
b
c
d
e
f
→
AT =
a
d
b
e
c
f

Example 1: A (2×3) → Aᵀ (3×2)

Step 1 - Write the matrix A.

In this problem: A has size 2×3.

A =
1
2
3
4
5
6

Step 2 - Swap rows and columns to form Aᵀ.

In this problem: AT will have size 3×2.

Swap rows and columns:
( AT )ij = aji

Step 3 - Read off the transposed matrix.

In this problem: Write the new matrix by swapping indices.

AT =
1
4
2
5
3
6

Final answer:

1
4
2
5
3
6

Example 2: A (3×2) → Aᵀ (2×3)

Step 1 - Write the matrix A.

In this problem: A has size 3×2.

A =
-2
0
7
3
1
-4

Step 2 - Swap rows and columns to form Aᵀ.

In this problem: AT will have size 2×3.

Swap rows and columns:
( AT )ij = aji

Step 3 - Read off the transposed matrix.

In this problem: Write the new matrix by swapping indices.

AT =
-2
7
1
0
3
-4

Final answer:

-2
7
1
0
3
-4
Matrix A Setup
Matrix A