The transpose of a matrix is a new matrix obtained by flipping the original matrix over its diagonal. This means that the rows of the original matrix become the columns of the transposed matrix and vice versa.
### Definition
If you have a matrix \( A \) with elements \( a_{ij} \), where \( i \) is the row index and \( j \) is the column index, the transpose of matrix \( A \), denoted as \( A^T \) or \( A' \), is defined as:
\[
(A^T)_{ij} =