Matrix Transpose Calculator

Set the matrix order and write down its values. The calculator will readily find its transpose.

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Matrix Transpose Calculator

Use this calculator to find the transpose of any matrix containing real or complex numbers. Below, we explain the concept of matrix transposition, its properties, and how to quickly compute it using this free tool.

What Is a Matrix Transpose?

Definition: The transpose of a matrix is created by flipping its rows into columns and its columns into rows.

Example:

$$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \quad \Rightarrow \quad A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} $$

Basic Method

If a matrix has size $m \times n$, its transpose will have size $n \times m$.
Rows of the original matrix become columns in the transposed matrix.

Key Properties of Matrix Transpose

Let A and B be matrices, and let x be a scalar:

$$ (A^T)^T = A $$
$$ (A + B)^T = A^T + B^T $$
$$ (AB)^T = B^T A^T $$
$$ (xA)^T = xA^T $$
$$ \det(A^T) = \det(A) $$
$$ (A^{-1})^T = (A^T)^{-1} $$
Eigenvalues of A and A^T are identical for square matrices.

How to Compute a Transpose

Example 1: 2×2 Matrix

$$ A = \begin{bmatrix} 6 & 7 \\ 2 & 5 \end{bmatrix} \quad \Rightarrow \quad A^T = \begin{bmatrix} 6 & 2 \\ 7 & 5 \end{bmatrix} $$

Example 2: 3×4 Matrix

$$ B = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \end{bmatrix} \quad \Rightarrow \quad B^T = \begin{bmatrix} 1 & 5 & 9 \\ 2 & 6 & 10 \\ 3 & 7 & 11 \\ 4 & 8 & 12 \end{bmatrix} $$

How Matrix Transpose Calculator Works

Select the number of rows and columns for your matrix.
Click Set Matrices to create the input grid.
Enter all matrix elements in the provided fields.
Click Calculate to get the transpose instantly.

FAQs

What happens when a matrix is multiplied by its transpose?

Multiplying a matrix by its transpose is often used to form symmetric matrices or calculate dot products. You can verify results using a matrix multiplication calculator.

Do matrices always commute with their transpose?

Not always. Only symmetric or certain special matrices commute with their transpose.

Why is the product AA^T symmetric?

Because transposing the product swaps the order: (AA^T)^T = (A^T)^TA^T = AA^T, which makes it symmetric.

What is the purpose of matrix transpose?

Transposing simplifies computations in linear algebra, including matrix multiplication, solving linear systems, and vector operations.

Can all matrices be transposed?

Yes, any matrix of any dimension can have its rows and columns interchanged.

Conclusion

The transpose is a fundamental operation in linear algebra used in mathematics, physics, and engineering. This free matrix transpose calculator makes the process quick, accurate, and easy to understand.