a.) Let A and B be $n$ x $n$ matrices. Prove that the matrix products $AB$ and $BA$ have the same eigenvalues.
b.) Prove that every eigenvalue of a matrix A is also an eigenvalue of its transpose $A^T$. Also, prove that if v is an eigenvector of A with eigenvalue $\lambda$ and w is an eigenvector of $A^T$ with a different eigenvalue $\mu \ne \lambda$, then v and w are orthogonal vectors with respect to the dot product.
For a, I know that if their eigenvalues are the same then their eigenvectors must relate too.
For the first part of b, is it similar to proving that the $det(A) =det (A^T)$? And, I do not know how to do the second part.