7. Recall that a nilpotent matrix is a nonzero square matrix A such that there exists a positive integer k where Ak = 0 (a) Prove that if A is an n x n nilpotent matrix, then A + I cannot have an eigenvalue of 0. (b) Find two square matrices of the same size, A and B such that A and B are both nilpotent, but A+ B and AB are not nilpotent.

Q: 5. Prove that a symmetric matrix P is positive definite iff all eigenvalues of P are positive.

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