Let's solve the problem step by step.

We are given:

1. A matrix $A = \begin{pmatrix} 2 & 5 \\ 3 & 8 \end{pmatrix}$
2. The inverse of matrix $A$ is $A^{-1} = \begin{pmatrix} 8 & -5 \\ -3 & 2 \end{pmatrix}$.
3. We need to find matrix $X$ such that $X \cdot A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ 1 & 0 \end{pmatrix}$.

Step 1: Multiplying both sides by $A^{-1}$

To isolate $X$, we can multiply both sides of the equation $X \cdot A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ 1 & 0 \end{pmatrix}$ by $A^{-1}$ from the right:

$X \cdot A \cdot A^{-1} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ 1 & 0 \end{pmatrix} \cdot A^{-1}$

Since $A \cdot A^{-1} = I$ (the identity matrix), this simplifies to:

$X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ 1 & 0 \end{pmatrix} \cdot A^{-1}$

Step 2: Performing the matrix multiplication

We need to multiply $\begin{pmatrix} 0 & 1 \\ 0 & 0 \\ 1 & 0 \end{pmatrix}$ by $\begin{pmatrix} 8 & -5 \\ -3 & 2 \end{pmatrix}$.

Performing the multiplication:

$X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 8 & -5 \\ -3 & 2 \end{pmatrix} = \begin{pmatrix} (0 \cdot 8 + 1 \cdot -3) & (0 \cdot -5 + 1 \cdot 2) \\ (0 \cdot 8 + 0 \cdot -3) & (0 \cdot -5 + 0 \cdot 2) \\ (1 \cdot 8 + 0 \cdot -3) & (1 \cdot -5 + 0 \cdot 2) \end{pmatrix}$

This simplifies to:

$X = \begin{pmatrix} -3 & 2 \\ 0 & 0 \\ 8 & -5 \end{pmatrix}$

Step 3: Comparing with the given options

The correct matrix $X$ is $\begin{pmatrix} -3 & 2 \\ 0 & 0 \\ 8 & -5 \end{pmatrix}$, which matches option D.

Final Answer: Option D

Would you like more details or have any questions?

Here are 5 related questions you might find useful:

1. How do you compute the inverse of a 2x2 matrix?
2. What properties does a matrix and its inverse have when multiplied?
3. What is the significance of multiplying by an inverse matrix in linear equations?
4. How can you verify if a matrix multiplication result is correct?
5. What types of matrices are not invertible?

Tip: When performing matrix multiplication, carefully follow the row-by-column method to avoid errors in the calculation!