Let's carefully re-evaluate the solution, step by step, to identify any potential errors.

Step 1: Finding the Inverse of Matrix $A$

Given: $A = \begin{bmatrix} -5 & 2 \\ -8 & -7 \end{bmatrix}$

First, compute the determinant of $A$: $\text{Det}(A) = (-5 \times -7) - (2 \times -8) = 35 - 16 = 19$

Next, compute the inverse of $A$: $A^{-1} = \frac{1}{\text{Det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \frac{1}{19} \begin{bmatrix} -7 & -2 \\ 8 & -5 \end{bmatrix} = \begin{bmatrix} -\frac{7}{19} & -\frac{2}{19} \\ \frac{8}{19} & -\frac{5}{19} \end{bmatrix}$

So, the correct inverse of $A$ should be: $A^{-1} = \begin{bmatrix} -\frac{7}{19} & -\frac{2}{19} \\ \frac{8}{19} & -\frac{5}{19} \end{bmatrix}$

Step 2: Solving $A \mathbf{x} = \mathbf{b}$

Given: $\mathbf{b} = \begin{bmatrix} -4 \\ 4 \end{bmatrix}$

We find $\mathbf{x}$ using: $\mathbf{x} = A^{-1} \mathbf{b} = \begin{bmatrix} -\frac{7}{19} & -\frac{2}{19} \\ \frac{8}{19} & -\frac{5}{19} \end{bmatrix} \begin{bmatrix} -4 \\ 4 \end{bmatrix}$

Calculate each element of $\mathbf{x}$:

1. First element $x_1$: $x_1 = \left(-\frac{7}{19} \times -4\right) + \left(-\frac{2}{19} \times 4\right) = \frac{28}{19} - \frac{8}{19} = \frac{20}{19}$

2. Second element $x_2$: $x_2 = \left(\frac{8}{19} \times -4\right) + \left(-\frac{5}{19} \times 4\right) = -\frac{32}{19} - \frac{20}{19} = -\frac{52}{19}$

Thus, the solution $\mathbf{x}$ is: $\mathbf{x} = \begin{bmatrix} \frac{20}{19} \\ -\frac{52}{19} \end{bmatrix}$

In simplified terms: $\mathbf{x} = \begin{bmatrix} \frac{20}{19} \\ -\frac{52}{19} \end{bmatrix}$

Summary

1. The correct inverse of $A$ is: $A^{-1} = \begin{bmatrix} -\frac{7}{19} & -\frac{2}{19} \\ \frac{8}{19} & -\frac{5}{19} \end{bmatrix}$

2. The correct solution for $\mathbf{x}$ is: $\mathbf{x} = \begin{bmatrix} \frac{20}{19} \\ -\frac{52}{19} \end{bmatrix}$

Would you like more details or have any questions?

Here are 5 questions you might want to ask:

1. How can I verify the correctness of a matrix inverse?
2. Can you explain the determinant calculation in more detail?
3. How does matrix multiplication work?
4. What are some common mistakes when solving matrix equations?
5. Can you show me how to solve another matrix equation?

Tip: Always use exact fractions rather than decimal approximations when solving matrix equations to avoid precision errors.