Matrix Transpose Calculator – Find Transpose of Any Matrix

Find the transpose of any matrix instantly with our free online matrix transpose calculator. Swap rows and columns of any size matrix with a single click.

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Enter matrix values (each row on a new line, values separated by spaces or commas)

Current matrix: 2x2 | Detected from your input

Understanding Matrix Transpose

The transpose of a matrix is created by flipping the matrix over its main diagonal – essentially swapping rows with columns. The first row becomes the first column, the second row becomes the second column, and so on. This simple operation has profound implications in linear algebra.

Transpose is denoted by a superscript T (A^T) or sometimes a prime (A'). For square matrices, the diagonal elements stay in place while everything else reflects across the diagonal. For rectangular matrices, the dimensions flip – an m x n matrix becomes n x m.

Transpose Properties

Basic Properties

(A^T)^T = A (double transpose returns original)
(A + B)^T = A^T + B^T
(cA)^T = c(A^T)
(AB)^T = B^T A^T (order reverses!)

Special Matrices

Symmetric: A^T = A
Skew-symmetric: A^T = -A
Orthogonal: A^T = A^-1
Diagonal matrices are symmetric

Worked Examples

Example 1: 2x2 Matrix

A = [[1, 2], [3, 4]]

A^T = [[1, 3], [2, 4]]

Row 1 [1,2] becomes Column 1

Row 2 [3,4] becomes Column 2

Example 2: Rectangular 2x3

A = [[1, 2, 3], [4, 5, 6]] (2x3)

A^T = [[1, 4], [2, 5], [3, 6]] (3x2)

Dimensions flip: 2x3 becomes 3x2

Example 3: Symmetric Matrix

A = [[1, 2, 3], [2, 4, 5], [3, 5, 6]]

A^T = [[1, 2, 3], [2, 4, 5], [3, 5, 6]] = A

Symmetric matrices equal their transpose!

Example 4: Product Transpose

(AB)^T = B^T A^T (not A^T B^T!)

The order reverses when transposing a product

Like removing shoes before socks

Quick Fact

Symmetric matrices (where A = A^T) are incredibly important in physics and engineering. They always have real eigenvalues and orthogonal eigenvectors. The stress tensor in mechanics, the covariance matrix in statistics, and the Hamiltonian in quantum mechanics are all symmetric.

Frequently Asked Questions

What happens to the diagonal during transpose?

Diagonal elements stay exactly where they are! Since they're at positions [i,i], swapping row and column indices doesn't move them. Only off-diagonal elements change positions.

Can any matrix be transposed?

Yes! Every matrix has a transpose, regardless of size or shape. Square or rectangular, real or complex – transpose is always defined.

What's a symmetric matrix?

A symmetric matrix equals its own transpose: A = A^T. This means the matrix is mirror-symmetric across its main diagonal. Distance matrices and correlation matrices are typically symmetric.

Why does (AB)^T = B^T A^T?

Think of it like getting dressed: you put on socks then shoes, but remove shoes then socks. Matrix operations reverse order when "undoing" – and transpose is a kind of reflection that reverses the operation order.

How is transpose used in practice?

Transpose appears everywhere: solving least squares problems (A^T A), computing dot products (x^T y), converting row vectors to column vectors, and in machine learning for weight updates and gradient calculations.

What's the transpose of a row vector?

A row vector (1 x n) transposes to a column vector (n x 1). This conversion is essential for matrix multiplication, since you can only multiply when inner dimensions match.

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