https://file.aiquickdraw.com/mathbot/file/17270652836017ji5ku20.jpg

The image shows three matrices for which the inverse must be found using the Gauss-Jordan method. Here are the matrices:

1. \[
\begin{pmatrix}
2 & 1 & -1 \\
0 & 2 & 1 \\
5 & 2 & -3
\end{pmatrix}
\]
2. \[
\begin{pmatrix}
0 & 1 & 2 \\
1 & 2 & 3 \\
3 & 1 & 1
\end{pmatrix}
\]
3. \[
\begin{pmatrix}
-1 & 2 & 2 \\
2 & -1 & 2 \\
2 & 2 & -1
\end{pmatrix}
\]

I'll go ahead and solve the first matrix using the Gauss-Jordan elimination method and explain the steps in detail. Then, I can proceed with the other matrices.

### Matrix 1:
\[
A = \begin{pmatrix}
2 & 1 & -1 \\
0 & 2 & 1 \\
5 & 2 & -3
\end{pmatrix}
\]

The steps to compute the inverse using Gauss-Jordan elimination are as follows:

1. **Form the augmented matrix** by appending the identity matrix to \( A \). This looks like:
   \[
   \left( \begin{array}{ccc|ccc}
   2 & 1 & -1 & 1 & 0 & 0 \\
   0 & 2 & 1 & 0 & 1 & 0 \\
   5 & 2 & -3 & 0 & 0 & 1
   \end{array} \right)
   \]
   
2. **Apply row operations** to reduce the left-hand matrix to the identity matrix, and the right-hand matrix will transform into the inverse matrix.

I'll now calculate the inverse of this matrix.The inverse of the first matrix \( A \) is:

\[
A^{-1} = \begin{pmatrix}
8 & -1 & -3 \\
-5 & 1 & 2 \\
10 & -1 & -4
\end{pmatrix}
\]

I will proceed with finding the inverse of the second matrix using the same Gauss-Jordan method. Let me now calculate that.The inverse of the second matrix \( B \) is:

\[
B^{-1} = \begin{pmatrix}
0.5 & -0.5 & 0.5 \\
-4 & 3 & -1 \\
2.5 & -1.5 & 0.5
\end{pmatrix}
\]

Finally, I will compute the inverse of the third matrix.The inverse of the third matrix \( C \) is:

\[
C^{-1} = \begin{pmatrix}
-0.111 & 0.222 & 0.222 \\
0.222 & -0.111 & 0.222 \\
0.222 & 0.222 & -0.111
\end{pmatrix}
\]

To summarize, the inverses of the three matrices are:

1. \( A^{-1} \):
   \[
   \begin{pmatrix}
   8 & -1 & -3 \\
   -5 & 1 & 2 \\
   10 & -1 & -4
   \end{pmatrix}
   \]
2. \( B^{-1} \):
   \[
   \begin{pmatrix}
   0.5 & -0.5 & 0.5 \\
   -4 & 3 & -1 \\
   2.5 & -1.5 & 0.5
   \end{pmatrix}
   \]
3. \( C^{-1} \):
   \[
   \begin{pmatrix}
   -0.111 & 0.222 & 0.222 \\
   0.222 & -0.111 & 0.222 \\
   0.222 & 0.222 & -0.111
   \end{pmatrix}
   \]

Would you like further clarification on any of the steps or methods used? Here are five follow-up questions:

1. How does the Gauss-Jordan method differ from other matrix inversion techniques?
2. Can every square matrix be inverted using the Gauss-Jordan method?
3. What are the potential issues with finding the inverse of large matrices?
4. How can matrix inverses be applied in solving systems of linear equations?
5. Can the inverse of a matrix always be computed numerically?

**Tip:** The determinant of a matrix can tell you if the matrix is invertible. If the determinant is zero, the matrix has no inverse.

Using Gauss-Jordan method, find the inverse of the following matrices: 
 1) [2 1 -1; 0 2 1; 5 2 -3] 
 2) [0 1 2; 1 2 3; 3 1 1] 
 3) [-1 2 2; 2 -1 2; 2 2 -1]

This problem demonstrates how to find the inverse of three 3x3 matrices using the Gauss-Jordan elimination method. The step-by-step process is explained, including row operations and matrix transformations. Ideal for university students studying linear algebra.

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