Transpose of a Matrix Calculator

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:

Example 1: Take the values below.

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

In this problem: Matrix A has size \(2 \times 3\), so \(A^{T}\) will have size \(3 \times 2\).

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.

Final answer:

Example 2: Take the values below.

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

In this problem: Matrix A has size \(3 \times 3\), so \(A^{T}\) will have size \(3 \times 3\).

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.

Final answer:

See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

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

Step 1

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

In this problem: Matrix A has size \(2 \times 3\), so \(A^{T}\) will have size \(3 \times 2\).

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

Step 2

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.

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

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

Step 1

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

In this problem: Matrix A has size \(3 \times 3\), so \(A^{T}\) will have size \(3 \times 3\).

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

Step 2

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.

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

Final Answer

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