https://file.aiquickdraw.com/mathbot/file/17276845570863e98efnp.jpg

The image provides a system of equations solved using matrix operations. Here's a breakdown of the solution:

### Key elements from the image:
- **Inverse Matrix Calculation**: The matrix \( A^{-1} \) is given as:
  \[
  A^{-1} = \frac{1}{22} 
  \begin{pmatrix}
  -18 & 6 & 4 \\
  13 & -8 & -9 \\
  1 & -4 & 1 
  \end{pmatrix}
  \]
  
- **Vector \( B \)**:
  \[
  B = 
  \begin{pmatrix}
  2 \\
  -7 \\
  14
  \end{pmatrix}
  \]

- **Solution for \( X \)**: Using the inverse matrix formula \( X = A^{-1} \cdot B \):
  \[
  X = \frac{1}{22} 
  \begin{pmatrix}
  -18 & 6 & 4 \\
  13 & -8 & -9 \\
  1 & -4 & 1 
  \end{pmatrix}
  \cdot
  \begin{pmatrix}
  2 \\
  -7 \\
  14
  \end{pmatrix}
  \]

### Matrix Multiplication Steps:
1. First row:
   \[
   (-18 \cdot 2) + (6 \cdot -7) + (4 \cdot 14) = -36 - 42 + 56 = -22
   \]
   
2. Second row:
   \[
   (13 \cdot 2) + (-8 \cdot -7) + (-9 \cdot 14) = 26 + 56 - 126 = -44
   \]

3. Third row:
   \[
   (1 \cdot 2) + (-4 \cdot -7) + (1 \cdot 14) = 2 + 28 + 14 = 44
   \]

Thus, after multiplication:
\[
X = \frac{1}{22} 
\begin{pmatrix}
-22 \\
-44 \\
44
\end{pmatrix}
=
\begin{pmatrix}
1 \\
2 \\
-2
\end{pmatrix}
\]

### Solution for Variables:
\[
x_1 = 1, \quad x_2 = 2, \quad x_3 = -2
\]

This concludes that the system of equations has the solution \( x_1 = 1 \), \( x_2 = 2 \), and \( x_3 = -2 \).

Would you like further details or have any questions about this solution?

Here are 5 related questions to explore further:
1. How do you calculate the inverse of a matrix?
2. What are the steps for multiplying matrices?
3. How can you verify if a matrix is invertible?
4. How does the determinant affect the invertibility of a matrix?
5. Can this method be used for larger matrices?

**Tip**: Always ensure that the matrix determinant is non-zero before attempting to find its inverse!

Solve the system of linear equations using the inverse matrix method. Matrix A and vector B are given.

Learn how to solve a system of linear equations using the inverse matrix method. This detailed step-by-step guide covers key linear algebra concepts such as matrix inversion and matrix multiplication. Suitable for students in Grades 10-12.

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