Matrix Transpose Calculator

About Matrix Transpose Calculator

The transpose swaps rows and columns: element (i,j) becomes element (j,i). (AB)ᵀ = BᵀAᵀ.

How to use this calculator

1. Enter the entries of your 2×2 or 3×3 matrix into the input boxes.
2. Check that each number is in the correct row and column before calculating.
3. Click the transpose button to swap rows with columns.
4. Read the output matrix, where the first row becomes the first column, the second row becomes the second column, and so on.

The formula explained

The transpose formula says that the entry in row \(i\), column \(j\) of the transposed matrix is the entry from row \(j\), column \(i\) of the original matrix: \(A^T_{ij} = A_{ji}\). This means every element moves to its mirrored position across the main diagonal.

• A = the original matrix
• \(A^T\) = the transpose of matrix A
• i = the row index in the transposed matrix
• j = the column index in the transposed matrix

Step by step method

1. Write the original matrix and label its rows and columns.
2. For each entry, switch its row and column position, so \(a_{12}\) becomes \(a_{21}\), \(a_{13}\) becomes \(a_{31}\), and so on.
3. Place all entries into their new positions to form the transposed matrix.

Worked example

Problem. Find the transpose of the 3×3 matrix \(\begin{pmatrix}1 & 4 & 7 \ 2 & 5 & 8 \ 3 & 6 & 9\end{pmatrix}\).

1. Start with the original matrix. The first row is \((1, 4, 7)\), the second row is \((2, 5, 8)\), and the third row is \((3, 6, 9)\).
2. Swap rows and columns. The first column of the transpose is \((1, 4, 7)\), the second column is \((2, 5, 8)\), and the third column is \((3, 6, 9)\).
3. Write the new matrix as \(\begin{pmatrix}1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9\end{pmatrix}\).

Tips and common mistakes

• The diagonal entries do not move, because they stay in the same row and column position.
• A common mistake is to reverse the order of entries in a row instead of swapping rows and columns.

Frequently asked questions