The transpose of a matrix is made by turning every row into a column without changing the entries themselves. This apparently simple operation matters in linear algebra, data tables, coordinate transformations, statistics, and computer code because it changes how dimensions line up for multiplication. The notation is usually (A^T), read “A transpose.”
Turning row positions into column positions
If (A) has (m) rows and (n) columns, then (A^T) has (n) rows and (m) columns. The entry that was in row (i), column (j), moves to row (j), column (i). Nothing is sorted, reflected by value, or recalculated. Only the address of each entry changes.
For example, a 2 by 3 matrix becomes a 3 by 2 matrix. Reading the original one row at a time and writing those rows vertically is a reliable hand method. The interactive view below makes the correspondence explicit: change any entry on the left and follow it to the matching position on the right.
Transpose a 2 × 3 matrix
Each row on the left becomes a column on the right.
Properties that prevent order mistakes
Transposing twice returns the starting matrix. A sum may be transposed term by term, and a scalar may pass through the operation. The product rule needs more care: the order reverses. In other words, the transpose of (AB) is (B^T A^T), not (A^T B^T). The reversal is what makes the dimensions compatible.
Suppose (A) is 2 by 3 and (B) is 3 by 4. Then (AB) is 2 by 4, so its transpose is 4 by 2. On the other side, (B^T) is 4 by 3 and (A^T) is 3 by 2; their product is also 4 by 2. Keeping the dimensions beside each factor is often enough to catch the wrong order.
Symmetric and skew-symmetric matrices
A square matrix is symmetric when its transpose equals itself. Its entries mirror across the main diagonal, so the two positions (a_{ij}) and (a_{ji}) match. A square matrix is skew-symmetric when its transpose equals its negative; in that case every main-diagonal entry must be zero over the real numbers.
Every square matrix can be separated into a symmetric part and a skew-symmetric part. This decomposition is useful in geometry, mechanics, optimization, and quadratic forms because the two parts have different algebraic behavior. Rectangular matrices have transposes, but they cannot be symmetric in the usual sense because equality requires the same order.
Worked example with a rectangular matrix
Take a matrix whose first row is 2, −1, 4 and whose second row is 0, 3, 5. The first row becomes the first column, and the second row becomes the second column. The resulting matrix has three rows and two columns.
Check the result by comparing addresses: the value 4 began in position (1,3), so it must finish in position (3,1); the value 3 moves from (2,2) to (2,2), remaining on the main diagonal. A second transpose reconstructs the original array. For the broader rules governing addition and multiplication, continue with matrices and matrix operations.
Where the transpose is used
In a spreadsheet-like data matrix, rows may represent observations and columns may represent variables. Transposing swaps those roles. In dot products and least-squares models, column vectors are transposed into row vectors so products have the intended dimensions. In a real-valued rotation matrix, the transpose is also the inverse, but that is a special property of orthogonal matrices, not a rule for every matrix.
