The transpose of a matrix is an operation that flips the matrix over its diagonal, switching the row and column indices of the matrix. If a matrix is represented as \( A \), its transpose is denoted by \( A^T \).
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad A^T = \begin{pmatrix} a & c \\ b & d \end{pmatrix} \]
This shows the transpose of a 2×2 matrix. The rows become columns, and the elements are flipped over the diagonal.
\[ \begin{bmatrix} a & b \\ c & d \\ u & v \end{bmatrix}^T = \begin{bmatrix} a & c & u \\ b & d & v \end{bmatrix} \]
In this 3×2 matrix, the transpose becomes a 2×3 matrix by turning rows into columns.