AlgebraFree

Matrix Transpose Calculator

Transpose a 3x3 matrix by swapping rows and columns.

Enter Values

a11

a12

a13

a21

a22

a23

a31

a32

a33

Result

Enter values above and click Calculate to see your result.

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Formula

Core Formula

(A^T)_{ij} = A_{ji}

How it works: Rows become columns and columns become rows.

Worked Example

Enter 3x3 values and compute transpose.

What Matrix Transpose Does

The transpose of a matrix flips it across its main diagonal. Row 1 becomes column 1, row 2 becomes column 2, and so on. Entry (i,j) in the original becomes entry (j,i) in the transpose.

Transposing twice returns the original matrix: (A^T)^T = A
The determinant is preserved: det(A^T) = det(A)
A matrix that equals its own transpose is called symmetric (A = A^T)
Transpose is used in the adjoint formula: adj(A) = cofactor(A) transposed

Transpose is one of the most common matrix operations, appearing in dot products, least squares regression, orthogonal transformations, and many other areas of mathematics.

You can also calculate changes using our Matrix Determinant Calculator, Adjoint Matrix Calculator, Cofactor Matrix Calculator or Matrix Rank Calculator.

Frequently Asked Questions

Does transpose preserve the determinant?

Yes. The determinant of the transpose always equals the determinant of the original matrix: det(A^T) = det(A).

What happens if I transpose twice?

You get the original matrix back. Transpose is its own inverse operation: (A^T)^T = A.

What is a symmetric matrix?

A symmetric matrix is one where A = A^T. The entries are mirrored across the main diagonal. Symmetric matrices have real eigenvalues and appear frequently in physics and statistics.

How is transpose related to the adjoint?

The adjoint (adjugate) of a matrix is the transpose of its cofactor matrix. So computing the adjoint requires a transpose step: adj(A) = C^T where C is the cofactor matrix.

Can I embed this Matrix Transpose Calculator on my website?

Yes. Click the "Embed" button at the top of this page to customize the size, colors, and theme, then copy the iframe code. Paste it into any HTML page, WordPress site, or CMS. It is completely free, requires no signup, and works on all devices. You can also visit our embed guide at calculory.com/services/embed-calculators for more details.

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