Answer:

Given that PQ = I, where P and Q are square matrices, we need to determine if zero is an eigenvalue of P and/or Q.

To determine if zero is an eigenvalue, we need to check if there exists a nonzero vector v such that Pv = 0v = 0.

Let's analyze each option:

a) P and not Q: This means zero is an eigenvalue of P but not of Q. If zero is an eigenvalue of P, then there exists a nonzero vector v such that Pv = 0v = 0. However, since PQ = I, we can multiply both sides of the equation by Q: Q(Pv) = Q(0v) = Q0 = 0. This implies that Qv = 0, which means zero is also an eigenvalue of Q. Therefore, option a) is incorrect.

b) Q but not P: This means zero is an eigenvalue of Q but not of P. Similarly to the previous case, if zero is an eigenvalue of Q, then there exists a nonzero vector v such that Qv = 0v = 0. Multiplying the equation PQ = I by v, we get (PQ)v = Iv = v. Since PQ = I, we have Pv = v. This contradicts the assumption that Qv = 0, as Pv would also be nonzero. Therefore, option b) is incorrect.

c) Both P and Q: This means zero is an eigenvalue of both P and Q. If zero is an eigenvalue of P, then there exists a nonzero vector v such that Pv = 0v = 0. Similarly, if zero is an eigenvalue of Q, then there exists a nonzero vector u such that Qu = 0u = 0. Now, let's consider the equation PQ = I. If we multiply both sides of the equation by u, we get (PQ)u = Iu = u. Since PQ = I, we have Pu = u. This implies that Pv = u, which contradicts the assumption that Pv = 0. Therefore, option c) is incorrect.

d) Neither P nor Q: This means zero is not an eigenvalue of either P or Q. This is the correct answer because if zero is not an eigenvalue, then there does not exist a nonzero vector such that Pv = 0v = 0 or Qv = 0v = 0, satisfying the given condition PQ = I. Therefore, option d) is correct.

In conclusion, the correct answer is option d) Neither P nor Q.