↻ Matrix Transpose Calculator
Transpose a matrix (flip rows and columns)
How to Use This Calculator
Set Matrix Dimensions
Choose the number of rows and columns.
Enter Matrix Elements
Input all elements of your matrix.
Calculate
Click to compute the transpose.
View Result
See the transposed matrix (rows become columns).
Formula
(Aᵀ)ᵢⱼ = Aⱼᵢ
Element (i,j) of transpose equals element (j,i) of original
Definition:
The transpose of a matrix A, denoted Aᵀ, is formed by flipping the matrix across its diagonal: rows become columns and columns become rows.
Example:
If A = [[1 2], [3 4], [5 6]], then Aᵀ = [[1 3 5], [2 4 6]]
Properties:
- (Aᵀ)ᵀ = A (double transpose is identity)
- (A + B)ᵀ = Aᵀ + Bᵀ
- (cA)ᵀ = cAᵀ for scalar c
- (AB)ᵀ = BᵀAᵀ (order reverses!)
- det(Aᵀ) = det(A)
About Matrix Transpose Calculator
The Matrix Transpose Calculator computes the transpose of a matrix by swapping rows and columns. The transpose operation is fundamental in linear algebra with applications in optimization, statistics, and computational mathematics.
When to Use This Calculator
- Dual Spaces: Work with row and column vectors
- Systems of Equations: Express systems in transposed form
- Orthogonality: AᵀA and AAᵀ matrices
- Statistics: Covariance matrices, least squares
- Optimization: Dual problems, Lagrangian
Why Use Our Calculator?
- ✅ Any Size: Works with any matrix dimensions
- ✅ Instant Result: Quick transpose computation
- ✅ Dimension Change: Shows how dimensions swap
- ✅ Educational: Helps understand transpose
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Transpose: Flip matrix across diagonal
- Symmetric: A = Aᵀ (matrix equals its transpose)
- Skew-Symmetric: A = -Aᵀ (negative transpose)
- Row/Column Swap: m×n becomes n×m
- Orthogonal: AᵀA = I for orthogonal matrices
Applications
Least Squares: Normal equations use AᵀA.
Quadratic Forms: xᵀAx for symmetric A.
Frequently Asked Questions
What happens to matrix dimensions under transpose?
Dimensions swap: an m×n matrix becomes n×m. For example, 3×5 becomes 5×3. Square matrices (n×n) stay n×n.
Is transpose its own inverse?
No, but (Aᵀ)ᵀ = A, so taking transpose twice returns the original matrix. Transpose is an involution (its own inverse in the sense that applying twice is identity).
Why does (AB)ᵀ = BᵀAᵀ have reverse order?
This is a fundamental property: the transpose of a product is the product of transposes in reverse order. (ABC)ᵀ = CᵀBᵀAᵀ, continuing the pattern.
What's a symmetric matrix?
A symmetric matrix equals its transpose: A = Aᵀ. For such matrices, aᵢⱼ = aⱼᵢ. Only square matrices can be symmetric.
Does transpose commute with addition and scalar multiplication?
Yes! (A + B)ᵀ = Aᵀ + Bᵀ and (cA)ᵀ = cAᵀ. Transpose distributes nicely over addition and scalar multiplication, but not over matrix multiplication (order reverses).