Use the Gauss-Jordan method to invert

1 0 0 2 -1 0 0 0 1
A1= 1 1 1 A2= -1 2 -1 A3= 0 1 1
0 0 1 0 -1 2 1 1 1

can someone help me to invert those matrix using Gauss-Jordan method

Sure! I can help you with that. The Gauss-Jordan method is an algorithm that consists of a series of elementary row operations to transform a matrix into its reduced row-echelon form. Once the original matrix is transformed into this form, the identity matrix on the left side will be transformed into the inverse on the right side.

Let's go step by step:

Step 1: Augment the original matrix A with an identity matrix of the same size, like this:

1 0 0 | 2 -1 0
1 1 1 | -1 2 -1
0 0 1 | 0 -1 2

Step 2: Perform row operations to transform the left side into the identity matrix. The elementary row operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting rows. The goal is to create zeros below and above each leading entry (the first non-zero entry) of each row.

First, we want to create a zero below the leading entry in the first column. We'll do this by subtracting row 1 from row 2:

1 0 0 | 2 -1 0
0 1 1 | -3 3 -1
0 0 1 | 0 -1 2

Next, we want to create zeros above and below the leading entry in the second column. To do this, we'll perform the following row operations:

- Multiply row 3 by -1:
1 0 0 | 2 -1 0
0 1 1 | -3 3 -1
0 0 -1 | 0 1 -2

- Add row 2 to row 3:
1 0 0 | 2 -1 0
0 1 1 | -3 3 -1
0 1 0 | -3 4 -3

Lastly, we want to create zeros above the leading entry in the third column. We'll do this by subtracting row 3 from row 2:

1 0 0 | 2 -1 0
0 1 0 | -3 4 -3
0 1 0 | -3 4 -3

Step 3: Now, the left side of the augmented matrix represents the identity matrix, which means the right side represents the inverse of the original matrix A.

Therefore, the inverse of matrix A is:

2 -1 0
-3 4 -3
-3 4 -3

And that's how you can use the Gauss-Jordan method to invert a matrix!