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To solve the matrix \( G \) using the **Gauss-Jordan elimination method**, we aim to transform the given matrix into its **reduced row echelon form (RREF)**.

Given matrix:
\[
G = \begin{bmatrix}
-4 & -1 & 2 \\
-5 & 2 & 2 \\
2 & -3 & -2
\end{bmatrix}
\]

Here are the steps for the Gauss-Jordan method:

### Step 1: Form the Augmented Matrix
Since we are solving the system represented by matrix \( G \), we start by writing the augmented matrix (if there's no given system, we assume a standard augmented matrix for identity):

\[
\left[ G | I \right] = 
\left[
\begin{array}{ccc|ccc}
-4 & -1 & 2  & 1 & 0 & 0 \\
-5 & 2  & 2  & 0 & 1 & 0 \\
2  & -3 & -2 & 0 & 0 & 1
\end{array}
\right]
\]

Now, I'll proceed step by step to perform row operations to bring it to the RREF form. Let me compute that.After applying the Gauss-Jordan elimination method, the matrix \( G \) has been transformed into its **reduced row echelon form (RREF)**. The result, along with the identity matrix on the right, is:

\[
\begin{bmatrix}
1 & 0 & 0 & 0.1  & -0.4 & -0.3 \\
0 & 1 & 0 & -0.3 & 0.2  & -0.1 \\
0 & 0 & 1 & 0.55 & -0.7 & -0.65
\end{bmatrix}
\]

Thus, the inverse of the matrix \( G \) is:
\[
G^{-1} = \begin{bmatrix}
0.1  & -0.4 & -0.3 \\
-0.3 & 0.2  & -0.1 \\
0.55 & -0.7 & -0.65
\end{bmatrix}
\]

Let me know if you need further details or explanations!

### Related questions:
1. How is the Gauss-Jordan method different from Gaussian elimination?
2. Can every matrix be inverted using Gauss-Jordan elimination?
3. What happens if a matrix has a determinant of zero?
4. How can we verify the result of a matrix inversion?
5. What are the practical applications of finding the inverse of a matrix?

### Tip:
When performing Gauss-Jordan elimination, always ensure that the leading coefficient (pivot) is non-zero to avoid division errors.

Learn how to use the Gauss-Jordan elimination method to find the inverse of a 3x3 matrix. The solution involves performing row operations to achieve the reduced row echelon form. Suitable for advanced high school or college-level students.

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