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Transpose of a Matrix — Definition, Formula & Examples

Transpose of a Matrix

A matrix which is formed by turning all the rows of a given matrix into columns and vice-versa. The transpose of matrix A is written AT.

 

Example showing matrix [1 2 3; 4 5 6] transposed to matrix A^T = [1 4; 2 5; 3 6]

 

 

See also

Inverse of a matrix

Key Formula

If A=[aij], then AT=[aji]\text{If } A = [a_{ij}], \text{ then } A^T = [a_{ji}]
Where:
  • AA = The original matrix
  • ATA^T = The transpose of matrix A
  • aija_{ij} = The element in row i and column j of the original matrix A
  • ajia_{ji} = The element in row j and column i, meaning the row and column indices are swapped

Worked Example

Problem: Find the transpose of the 2×3 matrix A.
Step 1: Write out the original matrix A.
A=[123456]A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}
Step 2: Take the first row of A and write it as the first column of the transpose.
Row 1 of A:[123]→Column 1 of AT:[123]\text{Row 1 of } A: \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \rightarrow \text{Column 1 of } A^T: \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}
Step 3: Take the second row of A and write it as the second column of the transpose.
Row 2 of A:[456]→Column 2 of AT:[456]\text{Row 2 of } A: \begin{bmatrix} 4 & 5 & 6 \end{bmatrix} \rightarrow \text{Column 2 of } A^T: \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}
Step 4: Combine the columns to form the transposed matrix. Notice the dimensions changed from 2×3 to 3×2.
AT=[142536]A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}
Answer: The transpose is AT=[142536]A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}, a 3×2 matrix.

Another Example

This example uses a square matrix instead of a rectangular one, and demonstrates the important property that the transpose of a transpose returns the original matrix.

Problem: Find the transpose of the 3×3 symmetric-looking matrix B and verify that (BT)T=B(B^T)^T = B.
Step 1: Write out the original square matrix B.
B=[270358149]B = \begin{bmatrix} 2 & 7 & 0 \\ 3 & 5 & 8 \\ 1 & 4 & 9 \end{bmatrix}
Step 2: Swap rows and columns: the element in position (i, j) moves to position (j, i). For example, b12=7b_{12} = 7 moves to position (2, 1), and b21=3b_{21} = 3 moves to position (1, 2).
BT=[231754089]B^T = \begin{bmatrix} 2 & 3 & 1 \\ 7 & 5 & 4 \\ 0 & 8 & 9 \end{bmatrix}
Step 3: Now transpose BTB^T again by swapping its rows and columns. The diagonal elements (2, 5, 9) stay in place.
(BT)T=[270358149]=B(B^T)^T = \begin{bmatrix} 2 & 7 & 0 \\ 3 & 5 & 8 \\ 1 & 4 & 9 \end{bmatrix} = B
Step 4: This confirms the property that transposing a matrix twice returns the original matrix: (BT)T=B(B^T)^T = B.
Answer: BT=[231754089]B^T = \begin{bmatrix} 2 & 3 & 1 \\ 7 & 5 & 4 \\ 0 & 8 & 9 \end{bmatrix}, and transposing again gives back BB.

Frequently Asked Questions

What is the difference between the transpose and the inverse of a matrix?
The transpose swaps rows and columns—every matrix has a transpose, regardless of its size. The inverse, denoted A−1A^{-1}, is a matrix such that A⋅A−1=IA \cdot A^{-1} = I (the identity matrix). Only square matrices with a nonzero determinant have an inverse. These are fundamentally different operations, though for orthogonal matrices the transpose happens to equal the inverse.
What happens to the dimensions of a matrix when you transpose it?
If the original matrix is m×nm \times n (m rows and n columns), its transpose is n×mn \times m (n rows and m columns). The dimensions are flipped. For a square matrix (m=nm = n), the dimensions stay the same.
What is a symmetric matrix and how does it relate to the transpose?
A symmetric matrix is a square matrix that equals its own transpose: A=ATA = A^T. This means the element in row ii, column jj is always equal to the element in row jj, column ii. Symmetric matrices appear frequently in physics and statistics, for example as covariance matrices.

Transpose of a Matrix vs. Inverse of a Matrix

Transpose of a MatrixInverse of a Matrix
DefinitionSwap rows and columns of the matrixFind a matrix that multiplies with A to give the identity matrix
NotationATA^TA−1A^{-1}
ExistenceExists for every matrix of any sizeExists only for square matrices with nonzero determinant
Dimensionsm×n matrix becomes n×mn×n matrix stays n×n
Key property(AT)T=A(A^T)^T = AA⋅A−1=IA \cdot A^{-1} = I

Why It Matters

You encounter the transpose in nearly every area that uses matrices—solving systems of linear equations, computing dot products, and working with data in statistics. In linear algebra courses, key results like the formula for least-squares regression (x^=(ATA)−1ATb\hat{x} = (A^T A)^{-1} A^T b) depend directly on the transpose. Understanding it is also essential for recognizing symmetric and orthogonal matrices, which have special properties that simplify many calculations in physics and engineering.

Common Mistakes

Mistake: Only swapping some elements instead of all off-diagonal elements.
Correction: Every element aija_{ij} must move to position (j,i)(j, i). A systematic approach is to rewrite each row of the original as a column of the transpose, working from the first row to the last.
Mistake: Forgetting to change the dimensions of the matrix after transposing.
Correction: If the original matrix is m×nm \times n, the transpose must be n×mn \times m. A 2×3 matrix becomes a 3×2 matrix. Only square matrices keep the same dimensions.

Related Terms