1. What is Matrix Transposition?

Matrix transposition is an operation that flips a matrix over its main diagonal, switching the row and column indices of the matrix. The transpose of matrix A is denoted as A^T.

2. How Does the Calculator Work?

The calculator uses the matrix transposition formula:

Where:

• \( A \) — Original matrix
• \( A^T \) — Transposed matrix
• \( i,j \) — Row and column indices

Explanation: The element at row i, column j in the original matrix becomes the element at row j, column i in the transposed matrix.

3. Importance of Matrix Transposition

Details: Matrix transposition is fundamental in linear algebra and has applications in solving systems of linear equations, computer graphics, statistics (covariance matrices), and machine learning.

4. Using the Calculator

Tips: Enter your matrix using commas to separate elements within a row and semicolons to separate different rows. For example, "1,2,3;4,5,6" represents a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What are the properties of matrix transposition?
A: Key properties include: (A^T)^T = A, (A+B)^T = A^T+B^T, (AB)^T = B^TA^T, and (cA)^T = cA^T.

Q2: What's the difference between transpose and inverse?
A: Transposition rearranges matrix elements, while inversion finds a matrix that when multiplied by the original gives the identity matrix. Not all matrices are invertible.

Q3: What is a symmetric matrix?
A: A matrix is symmetric if it equals its own transpose (A = A^T). Symmetric matrices have real eigenvalues and orthogonal eigenvectors.

Q4: How does transposition affect matrix dimensions?
A: An m×n matrix becomes an n×m matrix when transposed. Square matrices remain the same size.

Q5: What are practical applications of matrix transposition?
A: Used in solving linear systems, computer graphics transformations, covariance matrices in statistics, and various machine learning algorithms.