find the inverse of the following matrices if they exist.

[1 -2 3]
[3 1 0]
[-2 1 1]
the following represent a 3 x 3 matrices

To find the inverse of a matrix, you can follow these steps:

Step 1: Set up the augmented matrix.
Create an augmented matrix by placing the given matrix on the left and the identity matrix of the same size on the right. In this case, the given matrix is a 3x3 matrix, so the identity matrix would be:
[1 0 0]
[0 1 0]
[0 0 1]

Combine the two matrices to get:
[1 -2 3 | 1 0 0]
[3 1 0 | 0 1 0]
[-2 1 1 | 0 0 1]

Step 2: Use row operations to transform the left matrix into the identity matrix.
Apply row operations to the augmented matrix until the left matrix becomes the identity matrix. Perform the same row operations on the right matrix as well. The left matrix would become the identity matrix if it is possible to find the inverse.

Using Gaussian elimination, let's perform row operations to transform the left matrix into the identity matrix:

- Multiply Row 1 by 3 and add it to Row 2:
[1 -2 3 | 1 0 0]
[0 7 -9 | 3 1 0]
[-2 1 1 | 0 0 1]

- Multiply Row 1 by -2 and add it to Row 3:
[1 -2 3 | 1 0 0]
[0 7 -9 | 3 1 0]
[0 -3 7 | -2 0 1]

- Multiply Row 2 by 3/7 and add it to Row 1:
[1 0 -5 | 4/7 -2/7 0]
[0 7 -9 | 3 1 0]
[0 -3 7 | -2 0 1]

- Multiply Row 2 by 1/7:
[1 0 -5 | 4/7 -2/7 0]
[0 1 -9/7 | 3/7 1/7 0]
[0 -3 7 | -2 0 1]

- Multiply Row 2 by 3 and add it to Row 3:
[1 0 -5 | 4/7 -2/7 0]
[0 1 -9/7 | 3/7 1/7 0]
[0 0 8/7 | -1/7 3/7 1]

- Multiply Row 3 by 7/8:
[1 0 -5 | 4/7 -2/7 0]
[0 1 -9/7 | 3/7 1/7 0]
[0 0 1 | -1/8 3/8 7/8]

- Multiply Row 3 by 9/7 and add it to Row 2:
[1 0 -5 | 4/7 -2/7 0]
[0 1 0 | -6/28 27/28 9/28]
[0 0 1 | -1/8 3/8 7/8]

- Multiply Row 3 by 5 and add it to Row 1:
[1 0 0 | -3/28 17/28 25/28]
[0 1 0 | -6/28 27/28 9/28]
[0 0 1 | -1/8 3/8 7/8]

Step 3: Interpret the right matrix.
The right matrix, after performing the same row operations, would represent the inverse of the given matrix. The resulting right matrix is the inverse matrix.

The inverse of the given matrix is:
[-3/28 17/28 25/28]
[-6/28 27/28 9/28]
[-1/8 3/8 7/8]