Matrix Transpose Calculator
Calculate the transpose of any matrix by swapping rows and columns.
Calculate Matrix Transpose
Transpose Examples
2×3 Matrix
Original: 1,2,3;4,5,6
Transpose: 1,4;2,5;3,6
Size: 2×3 → 3×2
Square Matrix
Original: 1,2;3,4
Transpose: 1,3;2,4
Size: 2×2 → 2×2
Symmetric Matrix
Original: 1,2;2,3
Transpose: 1,2;2,3
Same as original
How to Calculate Matrix Transpose
Definition
The transpose of a matrix A, denoted A^T, is formed by swapping rows and columns.
Process
- Element A[i,j] becomes A^T[j,i]
- Rows become columns
- Columns become rows
Properties
- (A^T)^T = A
- (A + B)^T = A^T + B^T
- (AB)^T = B^T A^T
- (kA)^T = kA^T
- For square matrices: det(A^T) = det(A)
Special Cases
- Symmetric matrix: A^T = A
- Skew-symmetric matrix: A^T = -A
- Orthogonal matrix: A^T = A⁻¹