A similarity transformation is known to preserve which of the following?

A similarity transformation is a process in linear algebra where one matrix can be transformed into another while keeping certain properties unchanged. The correct answer relates to the preservation of both the trace and determinant of a matrix, which are significant characteristics when dealing with these transformations.

When a matrix ( A ) is similar to a matrix ( B ), represented as ( B = P^{-1}AP ) for some invertible matrix ( P ), the trace of the matrix, which is the sum of its eigenvalues, remains constant. This is because similarity transformations do not affect the eigenvalues themselves; they only change the basis in which the matrix is represented.

Similarly, the determinant of a matrix, which is also dependent on the eigenvalues, is preserved during similarity transformations. Specifically, the determinant of ( B ) can be computed as ( \text{det}(B) = \text{det}(P^{-1})\text{det}(A)\text{det}(P) ). Since ( \text{det}(P^{-1}) \text{det}(P) = 1 ), the determinant remains unchanged.

Other options, while related to matrices, do not fully represent properties preserved under similarity transformations. The rank of a matrix