Transpose of a Matrix Calculator

Published on: October 12 2025
Final Answer: Free • Full Steps: Pro

This Transpose of a Matrix Calculator reflects a matrix along its main diagonal and shows the working line by line. Every entry in row \(i\) and column \(j\) moves to row \(j\) and column \(i\), which turns the rows into columns. An \(m \times n\) matrix becomes \(n \times m\), so the shape changes unless it was square to begin with. Following the same pattern every time makes the transpose easy to check and repeat.

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Swap the rows and the columns.

Formula:

\((A^{T})_{ij} = a_{ji}\)
See Example 1 Hide Example 1

Problem

\(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Swap the rows and the columns.

Step 1

Step 1 - Set up the problem.

In this problem: Matrix A is \(2 \times 3\), so its transpose will be \(3 \times 2\).

\(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}\)

Step 2

Step 2 - Write the formula.

In this problem: Transposing reflects the matrix along its main diagonal, so the entry in row \(i\) and column \(j\) ends up in row \(j\) and column \(i\).

\((A^{T})_{ij} = a_{ji}\)

Step 3

Step 3 - Swap the rows and the columns.

In this problem: Taking each entry in turn: \(1\) moves from \((1, 1)\) to \((1, 1)\), \(2\) moves from \((1, 2)\) to \((2, 1)\), \(3\) moves from \((1, 3)\) to \((3, 1)\), \(4\) moves from \((2, 1)\) to \((1, 2)\), \(5\) moves from \((2, 2)\) to \((2, 2)\), \(6\) moves from \((2, 3)\) to \((3, 2)\).

\(A^{T} = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}\)

Final Answer

\(A^{T} = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}\)
See Example 2 Hide Example 2

Problem

\(A = \begin{pmatrix} 2 & -1 & 4 \\ 0 & 7 & 3 \\ -5 & 6 & 8 \end{pmatrix}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Swap the rows and the columns.

Step 1

Step 1 - Set up the problem.

In this problem: Matrix A is \(3 \times 3\), so its transpose will be \(3 \times 3\).

\(A = \begin{pmatrix} 2 & -1 & 4 \\ 0 & 7 & 3 \\ -5 & 6 & 8 \end{pmatrix}\)

Step 2

Step 2 - Write the formula.

In this problem: Transposing reflects the matrix along its main diagonal, so the entry in row \(i\) and column \(j\) ends up in row \(j\) and column \(i\).

\((A^{T})_{ij} = a_{ji}\)

Step 3

Step 3 - Swap the rows and the columns.

In this problem: Taking each entry in turn: \(2\) moves from \((1, 1)\) to \((1, 1)\), \(-1\) moves from \((1, 2)\) to \((2, 1)\), \(4\) moves from \((1, 3)\) to \((3, 1)\), \(0\) moves from \((2, 1)\) to \((1, 2)\), \(7\) moves from \((2, 2)\) to \((2, 2)\), \(3\) moves from \((2, 3)\) to \((3, 2)\), \(-5\) moves from \((3, 1)\) to \((1, 3)\), \(6\) moves from \((3, 2)\) to \((2, 3)\), \(8\) moves from \((3, 3)\) to \((3, 3)\).

\(A^{T} = \begin{pmatrix} 2 & 0 & -5 \\ -1 & 7 & 6 \\ 4 & 3 & 8 \end{pmatrix}\)

Final Answer

\(A^{T} = \begin{pmatrix} 2 & 0 & -5 \\ -1 & 7 & 6 \\ 4 & 3 & 8 \end{pmatrix}\)
Matrix A

Example Problems

Click one to load it into the box and solve it.