Calculate the transpose of any matrix instantly. Free matrix transpose calculator for 2×2, 3×3, 4×4, and larger matrices with step-by-step solutions.
Matrix Transpose Calculator
Visualize how Rows become Columns (AT)
Quick Test Example:
Original Matrix:
[1 2 3]
[4 5 6]Transpose Result:
[1 4]
[2 5]
[3 6]How to Use This Matrix Transpose Calculator
Step 1: Select Matrix Dimensions
Choose your matrix size from the options above (2×2, 3×3, 4×4, or enter custom dimensions). The calculator supports matrices up to 10×10.
Step 2: Enter Matrix Elements
Click on each cell and enter your numbers. You can use:
- Integers: 1, 2, 3, -5
- Decimals: 1.5, 2.75, -3.14
- Fractions: Will be converted to decimals
Step 3: Calculate the Transpose
Click the “Calculate Transpose” button to instantly compute the transpose of your matrix. The result shows the transposed matrix with all elements switched across the main diagonal.
Step 4: View Step-by-Step Solution
Enable “Show Steps” to see exactly how the transpose operation works, including which elements moved to which positions.
💡 Pro Tip: Use the “Quick Examples” buttons to load sample matrices and see how transpose works for different matrix types (identity matrix, symmetric matrix, diagonal matrix).
What is a Matrix Transpose?
The transpose of a matrix is an operation that flips a matrix over its main diagonal (the line from top-left to bottom-right). When you transpose a matrix A, you create a new matrix A^T where:
- Rows become columns
- Columns become rows
- The element at position (i,j) moves to position (j,i)
Mathematical Notation
If A is an m×n matrix, then A^T (A transpose) is an n×m matrix where:
A^T[i,j] = A[j,i]
Simple Example
For a 2×3 matrix:
Original Matrix A:
[1 2 3]
[4 5 6]
Transpose A^T:
[1 4]
[2 5]
[3 6]What happened?
- Row 1 [1, 2, 3] became Column 1
- Row 2 [4, 5, 6] became Column 2
Matrix Transpose Examples with Steps
Example 1: Transpose of a 2×2 Matrix
Given Matrix:
A = [3 7]
[2 5]Step-by-Step Solution:
Step 1: Identify the matrix dimensions
- Original: 2×2 (2 rows, 2 columns)
- Transpose will be: 2×2
Step 2: Apply transpose operation
- Element A[1,1] = 3 stays at position [1,1]
- Element A[1,2] = 7 moves to position [2,1]
- Element A[2,1] = 2 moves to position [1,2]
- Element A[2,2] = 5 stays at position [2,2]
Step 3: Write the transpose matrix
A^T = [3 2]
[7 5]Result: The transpose of matrix A is a 2×2 matrix where rows and columns are swapped.
Example 2: Transpose of a 3×3 Matrix
Given Matrix:
B = [1 2 3]
[4 5 6]
[7 8 9]Step-by-Step Solution:
Step 1: Original matrix is 3×3, transpose will also be 3×3
Step 2: Swap rows and columns
Row 1: [1, 2, 3] → Column 1 in B^T
Row 2: [4, 5, 6] → Column 2 in B^T
Row 3: [7, 8, 9] → Column 3 in B^TStep 3: Construct transpose matrix
B^T = [1 4 7]
[2 5 8]
[3 6 9]Verification:
- B[1,2] = 2, and B^T[2,1] = 2 ✓
- B[2,3] = 6, and B^T[3,2] = 6 ✓
- B[3,1] = 7, and B^T[1,3] = 7 ✓
Example 3: Transpose of a Rectangular Matrix (2×4)
Given Matrix:
C = [1 2 3 4]
[5 6 7 8]Step-by-Step Solution:
Step 1: Identify dimensions
- Original: 2×4 (2 rows, 4 columns)
- Transpose will be: 4×2 (4 rows, 2 columns)
Step 2: Convert each row to a column
Row 1: [1, 2, 3, 4] becomes Column 1
Row 2: [5, 6, 7, 8] becomes Column 2Step 3: Write transpose matrix
C^T = [1 5]
[2 6]
[3 7]
[4 8]Key Insight: Notice how a 2×4 matrix becomes a 4×2 matrix. The dimensions always flip when you transpose!
Example 4: Transpose with Negative Numbers
Given Matrix:
D = [-2 4 0]
[ 6 -1 3]Solution:
D^T = [-2 6]
[ 4 -1]
[ 0 3]Note: The transpose operation doesn’t change the values of elements, only their positions.
Related Matrix Calculators
Explore our other free matrix tools to complete your linear algebra calculations:
Core Matrix Operations
- Matrix Multiplication Calculator – Multiply two matrices with step-by-step solutions
- Matrix Inverse Calculator – Find inverse of 2×2, 3×3, 4×4 matrices
- Matrix Determinant Calculator – Calculate determinants with cofactor expansion
- Matrix Addition Calculator – Add and subtract matrices of any size
Advanced Matrix Tools
- RREF Calculator – Reduce matrices to row echelon form with steps
- Matrix Rank Calculator – Find the rank of any matrix
- Eigenvalue Calculator – Compute eigenvalues and eigenvectors
- Matrix Power Calculator – Calculate A^n for any power n
Vector Operations
- Dot Product Calculator – Calculate dot product of two vectors
- Cross Product Calculator – Find cross product of 3D vectors
- Vector Projection Calculator – Project one vector onto another
Specialized Calculators
- Orthogonal Matrix Calculator – Check if matrix is orthogonal
- Symmetric Matrix Calculator – Verify matrix symmetry
- Diagonal Matrix Calculator – Extract diagonal elements