what is transpose in a matrix. explain with the simplest example

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: \[ (A^T)_{ij} = A_{ji} \] ### Simple Example Let's consider a simple \( 2 \times 3 \) matrix \( A \): \[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \] Here, matrix \( A \) has 2 rows and 3 columns. To find the transpose \( A^T \), we swap the rows and columns: The first row of \( A \) (which is \( (1, 2, 3) \)) becomes the first column of \( A^T \). The second row of \( A \) (which is \( (4, 5, 6) \)) becomes the second column of \( A^T \). Thus, the transposed matrix \( A^T \) will be a \( 3 \times 2 \) matrix: \[ A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix} \] ### Summary In summary, the transposition of a matrix involves converting rows into columns and vice versa. In our example, a \( 2 \times 3 \) matrix became a \( 3 \times 2 \) matrix after transposition.