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Matrix Transpose

Matrix transpose calculator. Enter A to swap rows and columns into Aᵀ, with dimension notes—for matrix basics.

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(Aᵀ)ᵢⱼ = aⱼᵢ
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Matrix Transpose

1. Definition
Swapping rows and columns of A gives its transpose, denoted Aᵀ.
2. Symbols
SymbolMeaning
Aᵀtranspose of A
(i,j)original position
(j,i)transposed position
3. How this tool works
  • Transpose turns m×n into n×m;
  • Entry aᵢⱼ moves to position (j,i);
  • (Aᵀ)ᵀ = A.
4. Steps
  1. Enter A;
  2. Click Calculate to see Aᵀ;
  3. Note rows and columns swap.
5. Example
Example: A=1 2 3;4 5 6 (2×3), Aᵀ=1 4;2 5;3 6 (3×2).
6. Pitfalls
Transpose changes the size unless A is square;
A square matrix stays square;
The (i,j) entry moves to (j,i).

❓ FAQ｜Matrix Transpose

How is transpose defined?
Swap rows and columns: (Aᵀ)ᵢⱼ = aⱼᵢ.
How does the size change?
m×n becomes n×m.
What is (Aᵀ)ᵀ?
It equals A.
Any link to the inverse?
An orthogonal matrix satisfies Aᵀ=A⁻¹.
Does a square matrix stay square?
Yes, rows and columns stay equal.