Matrix Transpose Calculator – Swap Rows and Columns
Transpose a matrix by swapping rows and columns
Table of Contents
How to Use
- Select matrix size (2x2 or 3x3)
- Enter the matrix elements
- Click calculate to see the transposed matrix
- Review the original and transposed matrices side by side
What is a Matrix Transpose?
The transpose of a matrix is an operation that flips the matrix over its diagonal. This means the rows of the original matrix become the columns of the transposed matrix, and vice versa. If A is the original matrix, its transpose is denoted as Aᵀ or A'.
How Transposition Works
For any element at position (i, j) in the original matrix, it moves to position (j, i) in the transposed matrix. For example, the element in row 1, column 2 becomes the element in row 2, column 1.
Properties of Matrix Transpose
- Double transpose returns the original: (Aᵀ)ᵀ = A
- Transpose of sum: (A + B)ᵀ = Aᵀ + Bᵀ
- Transpose of product: (AB)ᵀ = BᵀAᵀ (note the reversed order)
- Transpose of scalar multiple: (cA)ᵀ = cAᵀ
- Symmetric matrices: A = Aᵀ
Applications
- Computing dot products and inner products
- Solving systems of linear equations
- Data science and machine learning
- Computer graphics transformations
- Signal processing and image manipulation
Frequently Asked Questions
- What happens to the dimensions when transposing?
- When you transpose a matrix, its dimensions are swapped. An m×n matrix becomes an n×m matrix. For square matrices (like 2x2 or 3x3), the dimensions remain the same.
- What is a symmetric matrix?
- A symmetric matrix is a square matrix that equals its own transpose (A = Aᵀ). This means the matrix is symmetric about its main diagonal, with elements mirrored across it.
- Why is the order reversed in (AB)ᵀ = BᵀAᵀ?
- The order reverses because of how matrix multiplication works. When you transpose a product, you must reverse the order of the factors and transpose each one individually.