Problem 1: Suppose A € M₂ is positive semi-definite and B ¤ M₂ is Hermitian. Prove that theeigenvalues of AB are real. Suppose A, B ¤ Mñ are positive semi-definite. Prove that eigenvaluesof AB are all nonnegative. Suppose A, B ¤ Mɲ are positive definite. Prove that eigenvalues ofAB are all positive.