About Matrix Transpose

The transpose of a matrix \( A \) (denoted \( A^T \)) swaps its rows and columns. For a matrix \( A \) of size \( m \times n \), \( A^T \) is \( n \times m \), where:

• \( (A^T)_{ij} = A_{ji} \)

Example: For \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} \), \( A^T = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix} \).