1. What is Matrix Transpose?

The transpose of a matrix is a new matrix whose rows are the columns of the original. If A is an m×n matrix, then A^T is an n×m matrix where \( A^T_{ij} = A_{ji} \).

2. How Does the Calculator Work?

The calculator uses the matrix transpose 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 transposed matrix equals the element at row j, column i in the original matrix.

3. Importance of Matrix Transpose

Details: Matrix transposition is fundamental in linear algebra, used in solving systems of equations, matrix decompositions, and various applications in physics, computer graphics, and statistics.

4. Using the Calculator

Tips: Enter matrix elements separated by commas for columns and semicolons for 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 transpose?
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: Transpose swaps rows and columns, while inverse (when it exists) is a matrix that when multiplied by the original gives the identity matrix.

Q3: What is a symmetric matrix?
A: A matrix that equals its transpose (A = A^T). These have important properties in linear algebra.

Q4: How does transpose affect matrix multiplication?
A: The transpose reverses the order of multiplication: (AB)^T = B^TA^T.

Q5: What are practical applications of matrix transpose?
A: Used in solving least squares problems, covariance matrices in statistics, image processing operations, and more.