Solve the system of linear equations  {█(2x+3y-z=11@-x+2y+3z=9@3x-2y+2z=16)┤  using either one of the method The inverse matrix method or Gauss method and check your result using cramer’s  rule.

Let's solve the system of linear equations using the **Inverse Matrix Method** first, and then check the solution using **Cramer's Rule**.

The given system of equations is:

\[
\begin{aligned}
1) &\ 2x + 3y - z = 11 \\
2) &\ -x + 2y + 3z = 9 \\
3) &\ 3x - 2y + 2z = 16
\end{aligned}
\]

### Step 1: Express the system in matrix form \( A \mathbf{x} = \mathbf{b} \)
The system can be written in the matrix form as:

\[
A = \begin{pmatrix}
2 & 3 & -1 \\
-1 & 2 & 3 \\
3 & -2 & 2
\end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 11 \\ 9 \\ 16 \end{pmatrix}
\]

Thus, the system becomes:

\[
A \mathbf{x} = \mathbf{b}
\]

### Step 2: Find the inverse of matrix \( A \)
To solve for \( \mathbf{x} \), we need to find the inverse of matrix \( A \), so that:

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

Let me calculate the inverse of matrix \( A \) and solve for \( \mathbf{x} \).The solution to the system of equations using the Inverse Matrix Method is approximately:

\[
x = 4.39, \quad y = 1.82, \quad z = 3.25
\]

### Step 3: Verify the result using Cramer's Rule
Now, let's check this result using **Cramer's Rule**. According to Cramer's Rule, the solution for \( x \), \( y \), and \( z \) can be found using the following determinants:

\[
x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)}, \quad z = \frac{\text{det}(A_z)}{\text{det}(A)}
\]

Where \( A_x \), \( A_y \), and \( A_z \) are matrices obtained by replacing the respective columns of \( A \) with the vector \( \mathbf{b} \).

Let me calculate the determinants and verify the result.Using **Cramer's Rule**, the solution is confirmed to be:

\[
x = 4.39, \quad y = 1.82, \quad z = 3.25
\]

This matches the result obtained using the **Inverse Matrix Method**, confirming the accuracy of the solution.

---

### Do you want further details or have any questions?

Here are 5 related questions to deepen understanding:

1. How do you calculate the inverse of a matrix by hand?
2. What conditions make Cramer's Rule applicable to a system of equations?
3. Can Cramer's Rule be applied to systems with dependent equations?
4. How does Gaussian elimination compare to the inverse matrix method?
5. What is the significance of the determinant in solving linear systems?

### Tip: When solving systems of equations, always check for the determinant first. If it's zero, the system has no unique solution or is inconsistent.

Solve the system of linear equations { 2x+3y-z=11, -x+2y+3z=9, 3x-2y+2z=16 } using either the inverse matrix method or Gauss method and check your result using Cramer's rule.

This guide explains how to solve a system of linear equations using the inverse matrix method and verifies the solution with Cramer's Rule. Key concepts covered include matrix inversion, determinants, and Cramer's Rule. Suitable for advanced high school or undergraduate students.

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