The transpose of a matrix is a new matrix whose rows are the columns of the original, and whose columns are the rows of the original. It is denoted as \(A^T\).

The Process:

To find the transpose, you simply "flip" the matrix over its main diagonal. The element at position \((i, j)\) in the original matrix moves to position \((j, i)\) in the transposed matrix.

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

Example:

If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), then \( A^T = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \)