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The image shows a series of matrix-related problems. Here is a breakdown of the content:

1. **Question 6**: It asks to determine the value of \( x \) given the matrix \( M = \begin{bmatrix} 9 & x \\ 8 & 7 \end{bmatrix} \) and \( \text{det}(M) = 9 \). This means that the determinant of matrix \( M \) equals 9, and the value of \( x \) must be found.

2. **Question 7**: It asks to find the inverse of matrix \( A = \begin{bmatrix} 3 & -7 \\ -1 & 2 \end{bmatrix} \).

3. **Question 8**: It asks to find the inverse of a matrix \( \begin{bmatrix} 1 & -3 & 1 \\ 2 & -2 & -1 \\ 3 & -1 & 1 \end{bmatrix} \).

Let's solve these problems step by step.

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### 6. Finding \( x \) from the determinant:
Given matrix \( M = \begin{bmatrix} 9 & x \\ 8 & 7 \end{bmatrix} \), the determinant is:
\[
\text{det}(M) = 9 \cdot 7 - 8 \cdot x = 9
\]
This simplifies to:
\[
63 - 8x = 9
\]
\[
8x = 63 - 9 = 54
\]
\[
x = \frac{54}{8} = 6.75
\]
Thus, \( x = 6.75 \).

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### 7. Finding the inverse of matrix \( A = \begin{bmatrix} 3 & -7 \\ -1 & 2 \end{bmatrix} \):
To find the inverse, we use the formula for a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), where the inverse is:
\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]
The determinant of \( A \) is:
\[
\text{det}(A) = (3)(2) - (-1)(-7) = 6 - 7 = -1
\]
So, the inverse is:
\[
A^{-1} = \frac{1}{-1} \begin{bmatrix} 2 & 7 \\ 1 & 3 \end{bmatrix} = \begin{bmatrix} -2 & -7 \\ -1 & -3 \end{bmatrix}
\]

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### 8. Finding the inverse of matrix \( \begin{bmatrix} 1 & -3 & 1 \\ 2 & -2 & -1 \\ 3 & -1 & 1 \end{bmatrix} \):
For a 3x3 matrix, the process of finding the inverse is more complex and involves calculating the determinant, then finding the matrix of minors, the cofactor matrix, and finally the adjugate matrix. Let me compute the inverse matrix for this system now.The inverse of the matrix \( \begin{bmatrix} 1 & -3 & 1 \\ 2 & -2 & -1 \\ 3 & -1 & 1 \end{bmatrix} \) is:
\[
\begin{bmatrix}
-0.1875 & 0.125 & 0.3125 \\
-0.3125 & -0.125 & 0.1875 \\
0.25 & -0.5 & 0.25
\end{bmatrix}
\]

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Would you like further details on any step? Here are some related questions:

1. How do you compute the determinant of a 3x3 matrix?
2. What are the properties of an inverse matrix?
3. Can all matrices have an inverse?
4. What is the formula for the inverse of a 2x2 matrix?
5. How can the inverse matrix be used to solve systems of linear equations?

**Tip:** The determinant plays a key role in determining whether a matrix has an inverse. If the determinant is zero, the matrix is singular and does not have an inverse.

The original question submitted by the user as text.

Explore solutions to advanced matrix problems, including calculating determinants and finding inverses of 2x2 and 3x3 matrices. Learn the key formulas and properties with detailed examples suitable for university level or advanced high school students.

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