Show that lambda is an eigenvalue of A iff it is an eigenval

Show that lambda is an eigenvalue of A iff it is an eigenvalue of A^T.

Solution

Consider the characteristic polynomial of A^T : |I A^T | = |(I A) ^T | = |I A| (since a matrix and its transpose have the same determinant). This result is the characteristic polynomial of A, so AT and A have the same characteristic polynomial, and hence they have the same eigenvalues.