Matrix Transpose Calculator
Transpose a matrix by swapping its rows and columns (A^T[i][j] = A[j][i]).
What Is the Transpose of a Matrix?
The transpose of matrix A (written A^T or A') is formed by swapping rows and columns. The element at position (i,j) in A moves to position (j,i) in A^T:
If A is an m×n matrix, then A^T is an n×m matrix — dimensions are swapped.
Worked Example
Find the transpose of the 2×3 matrix A:
[4, 5, 6] [2, 5]
[3, 6]
The 2×3 matrix becomes a 3×2 matrix. Row 1 [1,2,3] becomes column 1.
Key Properties of the Transpose
- Double transpose: (A^T)^T = A
- Addition: (A + B)^T = A^T + B^T
- Scalar: (kA)^T = k(A^T)
- Product (reversed!): (AB)^T = B^T × A^T
- Determinant: det(A^T) = det(A)
- Inverse: (A^T)^-1 = (A^-1)^T
Special Matrix Types Defined by Transpose
- Symmetric matrix: A = A^T (e.g., covariance matrices)
- Skew-symmetric: A = -A^T (diagonal entries must be 0)
- Orthogonal matrix: A^T = A^-1 (rotation/reflection matrices)
Real-World Applications
- Statistics: The formula for least squares regression is β = (X^T X)^-1 X^T y.
- Machine learning: Backpropagation uses transposes to propagate gradients through weight matrices.
- Computer graphics: Orthogonal transformation matrices have inverses equal to their transposes, making them efficient to invert.
- Signal processing: The DFT matrix and its transpose are used in fast Fourier transform algorithms.
Frequently Asked Questions
What is the transpose of a matrix?
A^T is obtained by flipping A over its main diagonal — rows become columns. If A is m×n, A^T is n×m.
Is the transpose of a square matrix always square?
Yes. An n×n matrix transposes to another n×n matrix. A 3×2 matrix transposes to a 2×3 matrix.
What is a symmetric matrix?
A square matrix A where A = A^T. This means A[i][j] = A[j][i] for all i,j. Covariance matrices are always symmetric.
What is the key property of (AB)^T?
(AB)^T = B^T × A^T — the order is reversed. This is analogous to the reverse rule for inverses: (AB)^-1 = B^-1 A^-1.