1. Use parity of powers for symmetric/skew-symmetric matrices

Let $S$ be symmetric, so $S^T=S$, and let $K$ be skew-symmetric, so $K^T=-K$.

Then:

• For any positive integer $n$,

  $$(S^n)^T=(S^T)^n=S^n$$

  so every power of a symmetric matrix is symmetric.

• For a skew-symmetric matrix $K$,

  $$(K^n)^T=(K^T)^n=(-K)^n=(-1)^n K^n.$$

  Hence:

  • if $n$ is even, $K^n$ is symmetric,
  • if $n$ is odd, $K^n$ is skew-symmetric.

So in this question:

• $X^4, Y^4$ are symmetric,
• $X^{23}, Y^{23}$ are skew-symmetric,
• $Y^3$ is skew-symmetric,
• $Z^4, Z^3$ are symmetric.

2. Useful transpose facts

• Sum of two skew-symmetric matrices is skew-symmetric.
• Sum of two symmetric matrices is symmetric.
• For matrices $A,B$, $$(AB)^T=B^T A^T.$$
• Therefore, for any $A,B$, $$(AB-BA)^T=B^T A^T-A^T B^T.$$

3. Check Option A: $Y^3Z^4-Z^4Y^3$

Here:

• $Y^3$ is skew-symmetric,
• $Z^4$ is symmetric.

Let

Then

Using $(Z^4)^T=Z^4$ and $(Y^3)^T=-Y^3$,

So $M$ is symmetric, not skew-symmetric.

Hence, A is not correct.

4. Check Option B: $X^{44}+Y^{44}$

Since $44$ is even,

• $X^{44}$ is symmetric,
• $Y^{44}$ is symmetric.

Therefore,

is symmetric, not necessarily skew-symmetric.

Hence, B is not correct.

5. Check Option C: $X^4Z^3-Z^3X^4$

Here:

• $X^4$ is symmetric,
• $Z^3$ is symmetric.

Let

Then

Since both are symmetric,

So $N$ is skew-symmetric.

Hence, C is correct.

6. Check Option D: $X^{23}+Y^{23}$

Since $23$ is odd,

• $X^{23}$ is skew-symmetric,
• $Y^{23}$ is skew-symmetric.

Their sum is also skew-symmetric because

Hence, D is correct.

7. Final answer

The skew-symmetric matrices are:

This matches the stored correct answer.