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 down Matrix A and check its size.
  2. Swap rows and columns to write the transpose Aᵀ.

Formula:

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

Example 1: Take the matrix below.

A =
1
2
3
4
5
6

Step 1 - Write down Matrix A and check its size.

In this problem: Matrix A has size 2×3, so AT will have size 3×2.

A =
1
2
3
4
5
6

Step 2 - Swap rows and columns to write the transpose Aᵀ.

In this problem: Each entry in position (i, j) moves to position (j, i), so rows become columns.

AT =
1
4
2
5
3
6

Final answer:

1
4
2
5
3
6

Example 2: Take the matrix below.

A =
2
-1
4
0
7
3
-5
6
8

Step 1 - Write down Matrix A and check its size.

In this problem: Matrix A has size 3×3, so AT will have size 3×3.

A =
2
-1
4
0
7
3
-5
6
8

Step 2 - Swap rows and columns to write the transpose Aᵀ.

In this problem: Each entry in position (i, j) moves to position (j, i), so rows become columns.

AT =
2
0
-5
-1
7
6
4
3
8

Final answer:

2
0
-5
-1
7
6
4
3
8
Matrix A Setup
Matrix A