The transpose of a matrix is defined as the operation that flips the matrix over its diagonal. This means that the rows of the original matrix become the columns of the transposed matrix. For example, if you have matrix ( A ) with elements ( a_{ij} ), then the transpose ( A^T ) will have elements ( a_{ji} ). So, the first row of matrix ( A ) becomes the first column of ( A^T ), the second row becomes the second column, and so forth.
This definition is fundamental in linear algebra because it preserves many properties of the matrix, such as the inner product of vectors and plays a critical role in solving systems of equations, computing matrix determinants, and understanding eigenvalues and eigenvectors. This operation is distinctly different from the other choices, which describe unrelated processes. For instance, adding elements or subtracting from the identity matrix does not yield the transpose. Similarly, multiplying by the inverse relates to matrix transformations rather than rearrangements of elements, which is what the transpose primarily involves.