Matrix Transpose
Compute the transpose of any matrix. Get instant step-by-step solutions — completely free, no signup required.
Free Matrix Transpose Calculator
Enter your problem below to get an instant step-by-step solution.
About Matrix Transpose
Compute the transpose of any matrix. Linear algebra and matrix operations are foundational to computer graphics, machine learning, and solving systems of equations. Understanding matrix manipulations is critical for advanced mathematics.
Our Matrix Transpose calculator handles every type of problem in this topic. Whether you are a student working through homework, preparing for an exam, or a professional needing a quick calculation, you can enter any matrix transpose expression and receive a complete, verified solution with every step shown.
Matrix Transpose Formula
(Aᵀ)ᵢⱼ = Aⱼᵢ
The transpose flips a matrix over its main diagonal: rows become columns and columns become rows. An m×n matrix becomes n×m. Transposes appear in dot products, least-squares regression and in the definition of symmetric matrices (A = Aᵀ).
Worked Example
Find the transpose of [[1, 2, 3], [4, 5, 6]]
- The original is 2×3, so the result is 3×2
- Row 1 (1, 2, 3) becomes column 1
- Row 2 (4, 5, 6) becomes column 2
Answer: [[1, 4], [2, 5], [3, 6]]
Common Mistakes to Avoid
- Keeping the original dimensions instead of swapping them.
- Reversing the order of elements within a row.
- Assuming (AB)ᵀ = AᵀBᵀ — it is BᵀAᵀ.
How to Use the Matrix Transpose Calculator
- 1
Input your matrix or vector values for the Matrix Transpose problem
- 2
Press Enter or click the Solve button to run the calculation
- 3
Review the step-by-step solution that appears below the input
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Check the breakdown of each operation to understand the method
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Try a different problem to practice and build your understanding
Frequently Asked Questions
What is the Matrix Transpose formula?
(Aᵀ)ᵢⱼ = Aⱼᵢ. The transpose flips a matrix over its main diagonal: rows become columns and columns become rows. An m×n matrix becomes n×m. Transposes appear in dot products, least-squares regression and in the definition of symmetric matrices (A = Aᵀ).
Can you show a worked Matrix Transpose example?
Find the transpose of [[1, 2, 3], [4, 5, 6]]: The original is 2×3, so the result is 3×2; Row 1 (1, 2, 3) becomes column 1; Row 2 (4, 5, 6) becomes column 2. Answer: [[1, 4], [2, 5], [3, 6]].
Is this Matrix Transpose calculator free to use?
Yes — the Matrix Transpose calculator on MathSimplify is completely free. There is no signup, no subscription, and no usage limit. Enter any problem and get the step-by-step solution instantly.
Can I use this for Matrix Transpose homework and exam practice?
Absolutely. The step-by-step breakdown is designed specifically for learning. You can use it to check your work, understand where you went wrong, or study the method before an exam. The calculator shows the full working, not just the final answer.