Let's carefully review the process and check for any potential errors. We already calculated the inverse and the solution to the matrix equation, but there seems to be an error.

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

Step 1: Calculating the Determinant of $A$

$\text{Det}(A) = (-5 \times -7) - (2 \times -8) = 35 - 16 = 19$

Step 2: Finding the Inverse of $A$

$A^{-1} = \frac{1}{19} \begin{bmatrix} -7 & -2 \\ 8 & -5 \end{bmatrix} = \begin{bmatrix} -\frac{7}{19} & -\frac{2}{19} \\ \frac{8}{19} & -\frac{5}{19} \end{bmatrix}$

Step 3: 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 $x_2$: $x_2 = \left(\frac{8}{19} \times -4\right) + \left(-\frac{5}{19} \times 4\right) = -\frac{32}{19} - \frac{20}{19} = -\frac{52}{19}$

Rechecking the Inverse Calculation:

$A^{-1} = \frac{1}{19} \begin{bmatrix} -7 & 2 \\ 8 & -5 \end{bmatrix} = \begin{bmatrix} -\frac{7}{19} & \frac{2}{19} \\ \frac{8}{19} & -\frac{5}{19} \end{bmatrix}$

There seems to have been a mistake in the signs in the inverse calculation provided in your entry. The correct inverse should be: $A^{-1} = \begin{bmatrix} -\frac{7}{19} & \frac{2}{19} \\ \frac{8}{19} & -\frac{5}{19} \end{bmatrix}$

Correct Solutions:

1. Inverse matrix: $A^{-1} = \begin{bmatrix} -\frac{7}{19} & \frac{2}{19} \\ \frac{8}{19} & -\frac{5}{19} \end{bmatrix}$

2. Solution to the matrix equation: $\mathbf{x} = \begin{bmatrix} \frac{20}{19} \\ -\frac{52}{19} \end{bmatrix}$

So, the corrected values should be:

$A^{-1} = \begin{bmatrix} -\frac{7}{19} & \frac{2}{19} \\ \frac{8}{19} & -\frac{5}{19} \end{bmatrix}$ $\mathbf{x} = \begin{bmatrix} \frac{20}{19} \\ -\frac{52}{19} \end{bmatrix}$

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 double-check the signs and elements when finding the inverse of a matrix, as small sign errors can lead to incorrect results.