Matrix Transpose Calculator

How it works

1. 1Choose the number of rows and columns (1–4 each) for matrix A using the selectors.
2. 2Fill in the cell values — blank or non-numeric cells are treated as 0.
3. 3The transposed matrix Aᵀ appears instantly: each element (i, j) moves to position (j, i), turning an m × n matrix into an n × m matrix.

Use cases

• Converting a row vector into a column vector before a matrix multiplication.
• Computing AᵀA or AAᵀ as part of least-squares regression or PCA.
• Verifying that a symmetric matrix satisfies A = Aᵀ.

Frequently asked questions

What does transposing a matrix mean?

Transposing flips a matrix over its main diagonal. Every element at position (i, j) moves to position (j, i), so the rows of A become the columns of Aᵀ. A 2 × 3 matrix becomes a 3 × 2 matrix after transposition.

Is the transpose of a symmetric matrix the same as the original?

Yes. A matrix is symmetric when A = Aᵀ, meaning element (i, j) equals element (j, i) for all positions. The transpose of a symmetric matrix is identical to the original.

How is the transpose used in real calculations?

The transpose appears throughout linear algebra: dot products via vᵀw, the normal equations AᵀAx = Aᵀb in least-squares fitting, covariance matrices, and checking orthogonality (where Aᵀ = A⁻¹).