A positive definite matrix must be a symmetric matrix, but a matrix similar to a symmetric matrix is not necessarily a symmetric matrix, so B is not necessarily symmetric, let alone a positive definite matrix. Options (a) and (d) are incorrect.

A itself is not necessarily orthogonal, and the similarity of two matrices has nothing to do with orthogonality or not. Option (b) is also wrong.

The necessary and sufficient condition for a matrix to be a positive definite matrix is that the eigenvalues are all positive, while similar matrices have the same eigenvalues, so the eigenvalues of b are all positive, so |B|= eigenvalue product > 0, and b must be reversible. Option (c) is correct.

The economic mathematics team will answer for you. Please comment in time if you are satisfied. thank you