The transpose operation reflects a matrix across its main diagonal. It preserves diagonal entries and swaps each off-diagonal pair across the diagonal.
⚠Transpose works for rectangular matrices too, but this page focuses on 2x2 and 3x3 square matrices to match the site matrix style.
What Is Matrix Transpose?
The transpose of a matrix changes the orientation of its entries by interchanging rows and columns. It is one of the most common basic operations in linear algebra.
Swap Indexes
Entry a_ij moves to position a_ji.
Diagonal Fixed
Main diagonal entries remain unchanged.
Symmetry Check
A equals A^T exactly when A is symmetric.
Common Operation
Transpose appears in products, projections, and data matrices.
Example: For A = [[1,2],[3,4]], the transpose is A^T = [[1,3],[2,4]].
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