Answer:

In order to determine which matrix, either PQ or QP, is Hermitian, we need to understand the properties of Hermitian and skew-Hermitian matrices.

A matrix P is said to be Hermitian if it is equal to its conjugate transpose. In other words, the elements of P satisfy the condition Pij = conj(Pji), where conj() denotes the complex conjugate.

On the other hand, a matrix Q is said to be skew-Hermitian if it is equal to the negative of its conjugate transpose. In other words, the elements of Q satisfy the condition Qij = -conj(Qji).

Now, let's analyze the product of the two matrices PQ and QP to determine which one is Hermitian.

Analysis:

1. PQ:
- The product of a Hermitian matrix (P) and a skew-Hermitian matrix (Q) may or may not be Hermitian.
- Let's consider the elements of the product PQ: (PQ)ij = ∑(Pkj * Qki), where k is the summation index.
- Taking the complex conjugate of (PQ)ij, we get (PQ)ji = ∑(conjugate(Pki) * conjugate(Qkj)).
- For PQ to be Hermitian, we need (PQ)ij = conjugate((PQ)ji).
- By substituting the expressions for (PQ)ij and (PQ)ji, we have ∑(Pkj * Qki) = ∑(conjugate(Pki) * conjugate(Qkj)).
- This equation does not hold in general, so PQ is not necessarily Hermitian.

2. QP:
- The product of a skew-Hermitian matrix (Q) and a Hermitian matrix (P) may or may not be Hermitian.
- Let's consider the elements of the product QP: (QP)ij = ∑(Qik * Pkj), where k is the summation index.
- Taking the complex conjugate of (QP)ij, we get (QP)ji = ∑(conjugate(Qki) * conjugate(Pkj)).
- For QP to be Hermitian, we need (QP)ij = conjugate((QP)ji).
- By substituting the expressions for (QP)ij and (QP)ji, we have ∑(Qik * Pkj) = ∑(conjugate(Qki) * conjugate(Pkj)).
- This equation does hold in general because the order of multiplication does not affect complex conjugation, so QP is Hermitian.

Conclusion:

In conclusion, the matrix QP is Hermitian.