The image presents a problem about matrices $A$ and $B$. The matrices are given as follows:

$A = \begin{bmatrix} 2 & -4 \\ 3 & -5 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$

The task is to determine several matrix expressions and their inverses:

1. $AB$
2. $BA$
3. $(AB)^{-1}$
4. $(BA)^{-1}$
5. $A^{-1}$
6. $B^{-1}$
7. $A^{-1}B$
8. $B^{-1}A^{-1}$

Let's go through the steps to compute these.

1. $AB$

We need to multiply matrices $A$ and $B$.

$AB = \begin{bmatrix} 2 & -4 \\ 3 & -5 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$

2. $BA$

Next, we calculate $BA$:

$BA = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \times \begin{bmatrix} 2 & -4 \\ 3 & -5 \end{bmatrix}$

3. $(AB)^{-1}$

Once we have the product $AB$, we can calculate the inverse $(AB)^{-1}$.

4. $(BA)^{-1}$

Similarly, after finding $BA$, we can compute $(BA)^{-1}$.

5. $A^{-1}$

We also need to calculate the inverse of matrix $A$, which involves finding the determinant and the adjugate matrix.

6. $B^{-1}$

Likewise, for matrix $B$, we'll calculate its inverse.

7. $A^{-1}B$

Using $A^{-1}$ and $B$, we compute $A^{-1}B$.

8. $B^{-1}A^{-1}$

Finally, we compute $B^{-1}A^{-1}$.

Let me calculate each of these step-by-step. I'll begin with the matrix multiplications.The matrix products are:

$AB = \begin{bmatrix} -6 & -8 \\ -7 & -9 \end{bmatrix}$

$BA = \begin{bmatrix} 8 & -14 \\ 13 & -23 \end{bmatrix}$

Next, I'll calculate the inverses of these matrices, along with the inverses of $A$ and $B$.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the steps to calculate the matrix inverses manually, or would you like to try solving it yourself?

Here are five related questions to expand your understanding:

1. How do you find the determinant of a 2x2 matrix?
2. What conditions make a matrix invertible?
3. How do you compute the inverse of a matrix manually?
4. What is the significance of multiplying matrices in different orders (e.g., $AB$ vs. $BA$)?
5. How do matrix inverses relate to solving systems of linear equations?

Tip: For a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, its inverse (if it exists) is given by $A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$, provided that $ad - bc \neq 0$.