The problem asks you to find the matrix $A$, which is given as the product of two block matrices:

$A = \begin{pmatrix} Z & 0 \\ X & Y \end{pmatrix} \begin{pmatrix} Y & X \\ -X & 0 \end{pmatrix}$

We will perform the multiplication of these two block matrices.

Block Matrix Multiplication:

We multiply the matrices block by block, using the rule:

$A = \begin{pmatrix} Z & 0 \\ X & Y \end{pmatrix} \begin{pmatrix} Y & X \\ -X & 0 \end{pmatrix}$ will result in:

$A = \begin{pmatrix} ZY + 0 \cdot (-X) & ZX + 0 \cdot 0 \\ XY + Y \cdot (-X) & XX + Y \cdot 0 \end{pmatrix}$

Thus, simplifying each element of the resulting matrix:

1. Top-left element: $ZY$
2. Top-right element: $ZX$
3. Bottom-left element: $XY - YX$ (note that $XY - YX$ is the commutator of $X$ and $Y$, which may not be zero since matrix multiplication is not commutative)
4. Bottom-right element: $XX$

Thus, the resulting matrix is:

$A = \begin{pmatrix} ZY & ZX \\ XY - YX & XX \end{pmatrix}$

Now, let's check the options:

• Option A: $A = \begin{pmatrix} YZ & ZX \\ 0 & XX \end{pmatrix}$

  • This is incorrect because the top-left element should be $ZY$, not $YZ$, and the bottom-left element is $XY - YX$, not 0.

• Option B: $A = \begin{pmatrix} ZY & ZX \\ 0 & XX \end{pmatrix}$

  • This is incorrect because the bottom-left element is $XY - YX$, not 0.

• Option C: $A = \begin{pmatrix} ZY & ZX \\ XY - YX & XX \end{pmatrix}$

  • This is correct. The elements match the product we calculated.

• Option D: None of the above

  • This is incorrect because Option C is correct.

Conclusion:

The correct answer is C.

Would you like further clarification on any of the steps? Here are 5 related questions that could expand your understanding:

1. What are some properties of block matrix multiplication?
2. How do commutators like $XY - YX$ appear in other areas of matrix algebra?
3. What are some conditions under which matrix multiplication is commutative?
4. How do zero matrices influence the product in block matrix multiplication?
5. Can the result change if the matrices $X, Y, Z$ have specific properties like being symmetric?

Tip:

When multiplying block matrices, treat each block like a regular matrix multiplication and ensure all dimensions align properly for valid operations.