**same for** 
**|1  1  1|**
**|0  2  3|**
**|5  5  1|**

Let's find the inverse of the given matrix using the Gauss-Jordan method. The matrix is:

\[
\mathbf{A} = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 3 \\ 5 & 5 & 1 \end{bmatrix}
\]

**Steps:**

1. **Augment the matrix** with the identity matrix:
   \[
   \left[\begin{array}{ccc|ccc}
   1 & 1 & 1 & 1 & 0 & 0 \\
   0 & 2 & 3 & 0 & 1 & 0 \\
   5 & 5 & 1 & 0 & 0 & 1
   \end{array}\right]
   \]

2. **Perform row operations** to convert the left side into the identity matrix. I will perform these steps now.The inverse of the given matrix using the Gauss-Jordan method is:

\[
\mathbf{A}^{-1} = \begin{bmatrix} 1.625 & -0.5 & -0.125 \\ -1.875 & 0.5 & 0.375 \\ 1.25 & 0 & -0.25 \end{bmatrix}
\]

### Steps Recap:
1. The augmented matrix \([A | I]\) was transformed using row operations.
2. The left side was converted to the identity matrix, resulting in the inverse matrix on the right.

Would you like a detailed explanation of each row operation, or is there something else you'd like to explore?

### Related Questions:
1. How can Gauss-Jordan elimination be performed manually?
2. What are the applications of the inverse of a matrix?
3. How to check the accuracy of the inverse matrix?
4. What is the relationship between matrix determinants and inverses?
5. How does the Gauss-Jordan method compare to other methods for finding inverses?

**Tip:** Always verify your result by multiplying the matrix by its inverse; the product should be the identity matrix.

Learn how to find the inverse of matrix [1 1 1; 0 2 3; 5 5 1] using the Gauss-Jordan method. Detailed step-by-step solution with augmented matrix transformations and row operations provided. Suitable for advanced high school students and anyone interested in matrix algebra.

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