The formula to calculate the transpose of a matrix is:
\[ A^T = [a_{ij}]^T = [a_{ji}] \]
where \( A \) is the original matrix, \( A^T \) is the transpose of the matrix, \( a_{ij} \) is the element in the \( i \)-th row and \( j \)-th column of the original matrix, and \( a_{ji} \) is the element in the \( j \)-th row and \( i \)-th column of the transpose matrix.
A matrix transpose is a fundamental operation in linear algebra which involves flipping a matrix over its diagonal, which starts from the top-left to the bottom-right, thus switching the row and column indices of each element. This means that the element at the \( i \)-th row and \( j \)-th column in the original matrix will be placed at the \( j \)-th row and \( i \)-th column in the transposed matrix. For instance, if the original matrix is denoted by \( A \), its transpose is usually denoted by \( A^T \) or \( A' \). The transpose of a matrix is used in a wide range of mathematical and scientific applications, including simplifying calculations in matrix algebra, solving systems of linear equations, and performing operations in computer graphics.
Let's assume we have the following 2x3 matrix:
1, 2, 3 4, 5, 6
Step 1: Interchange the rows and columns:
1, 4 2, 5 3, 6
Therefore, the transpose of the matrix is:
1, 4 2, 5 3, 6