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To solve for the inverse of the given matrix \( A \) using **Gauss-Jordan elimination**, follow these steps:

The matrix \( A \) is:

\[
A = \begin{pmatrix}
-2 & 2 & -1 \\
3 & -5 & 4 \\
5 & -6 & 4
\end{pmatrix}
\]

The inverse of matrix \( A \), if it exists, can be found by augmenting the matrix \( A \) with the identity matrix \( I \):

\[
[A | I] = \left( \begin{array}{ccc|ccc}
-2 & 2 & -1 & 1 & 0 & 0 \\
3 & -5 & 4 & 0 & 1 & 0 \\
5 & -6 & 4 & 0 & 0 & 1 \\
\end{array} \right)
\]

### Steps of Gauss-Jordan Elimination:
1. Convert the matrix \( A \) to reduced row echelon form (RREF).
2. Apply row operations to the augmented matrix until the left side becomes the identity matrix.
3. The right side will become the inverse of \( A \).

I'll perform the Gauss-Jordan elimination process to find the inverse of this matrix.The inverse of the matrix \( A \) using Gauss-Jordan elimination is:

\[
A^{-1} = \begin{pmatrix}
4 & -2 & 3 \\
8 & -3 & 5 \\
7 & -2 & 4
\end{pmatrix}
\]

This is the result of converting the augmented matrix into reduced row echelon form.

Would you like a detailed explanation of any of the steps, or do you have any other questions?

Here are 5 related questions to expand your understanding:
1. How does Gauss-Jordan elimination differ from Gaussian elimination?
2. What conditions must be satisfied for a matrix to have an inverse?
3. Can all square matrices be inverted using Gauss-Jordan elimination?
4. What are the applications of the inverse matrix in solving systems of equations?
5. How can you check if a matrix is singular before applying Gauss-Jordan elimination?

**Tip:** Always ensure the determinant of the matrix is non-zero before attempting to find its inverse. This is a quick way to check if the matrix is invertible.

Find the inverse of the following matrix by using Gauss-Jordan elimination A = \begin{pmatrix} -2 & 2 & -1 \\ 3 & -5 & 4 \\ 5 & -6 & 4 \end{pmatrix}.

Learn how to find the inverse of a matrix using Gauss-Jordan elimination. This problem involves transforming the matrix A = \begin{pmatrix} -2 & 2 & -1 \\ 3 & -5 & 4 \\ 5 & -6 & 4 \end{pmatrix} into its inverse using row reduction techniques. Suitable for undergraduate students studying linear algebra.

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