https://file.aiquickdraw.com/mathbot/file/1724459396256r64vrcvm.jpg

To solve part (b) from the provided image, let's analyze the problem:

The question seems to involve solving a system of linear equations, given in matrix form. The system is represented as \( Ax = b \), where \( A \) is a matrix of coefficients, \( x \) is a vector of variables, and \( b \) is a vector of constants.

In part (b), the task usually involves solving the matrix equation by finding the inverse of matrix \( A \) and then multiplying it by vector \( b \) to find the vector \( x \), using the relation:

\[
x = A^{-1}b
\]

Let me transcribe the matrices and vectors from the image, solve for \( x \), and provide the solution.

---

### Part (b) - Solution Steps:

1. **Given Matrix \( A \) and Vector \( b \):**
   \[
   A = \begin{pmatrix}
   3 & -2 & 1 \\
   2 & 1 & -3 \\
   1 & -1 & 2
   \end{pmatrix},
   \quad
   b = \begin{pmatrix}
   -1 \\
   9 \\
   -7
   \end{pmatrix}
   \]

2. **Find the inverse of matrix \( A \):**
   To compute the inverse, \( A^{-1} \), we can use the formula:
   \[
   A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
   \]
   where \( \text{det}(A) \) is the determinant of \( A \) and \( \text{adj}(A) \) is the adjugate of \( A \).

3. **Compute the determinant \( \text{det}(A) \):**
   \[
   \text{det}(A) = 3 \cdot (1 \cdot 2 - (-3) \cdot (-1)) - (-2) \cdot (2 \cdot 2 - (-3) \cdot 1) + 1 \cdot (2 \cdot (-1) - 1 \cdot 1)
   \]
   Simplify this to get the value of \( \text{det}(A) \).

4. **Find the adjugate matrix \( \text{adj}(A) \).**

5. **Calculate \( A^{-1} \).**

6. **Finally, multiply \( A^{-1} \) by \( b \) to find \( x \):**
   \[
   x = A^{-1} \cdot b
   \]

I will now carry out these calculations for you.The solution to the system of equations, represented by the vector \( x \), is:

\[
x = \begin{pmatrix}
-0.875 \\
-3.125 \\
-4.625
\end{pmatrix}
\]

This means the values of the variables corresponding to this vector are approximately:

- \( x_1 = -0.875 \)
- \( x_2 = -3.125 \)
- \( x_3 = -4.625 \)

Would you like further details on the steps, or do you have any other questions?

### Relative Questions:
1. How is the inverse of a matrix calculated step by step?
2. What is the significance of the determinant in matrix inversion?
3. How can systems of linear equations be solved using different methods like Gaussian elimination?
4. What is an adjugate matrix, and how is it computed?
5. How do numerical approximations affect the accuracy of solutions in matrix algebra?

### Tip:
When solving systems of linear equations, always check the determinant of the coefficient matrix. If the determinant is zero, the matrix is singular, and an inverse does not exist, indicating either no solution or infinitely many solutions.

Learn how to solve a system of linear equations using matrix inversion. This detailed solution involves finding the inverse of a matrix, calculating determinants, and applying matrix multiplication to find variable values. Suitable for advanced high school and college-level students.

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