To proceed with finding the inverse of the given matrix, we perform a series of row operations as shown in the problem.

Explanation of Each Step

1. Original Matrix Setup:
   The matrix provided is:

   3 & -3 & 6 \\ 1 & -3 & 10 \\ -1 & 3 & -5 \end{bmatrix}$$ This matrix is augmented with the identity matrix, resulting in: $$\begin{bmatrix} 3 & -3 & 6 & | & 1 & 0 & 0 \\ 1 & -3 & 10 & | & 0 & 1 & 0 \\ -1 & 3 & -5 & | & 0 & 0 & 1 \end{bmatrix}$$

2. Step 1: Row Swap
   The first row operation indicated is a swap between Row 1 and Row 2. This is done to bring the leading 1 to the top left position, making it easier to perform further row operations. After swapping, we get:

   1 & -3 & 10 & | & 0 & 1 & 0 \\ 3 & -3 & 6 & | & 1 & 0 & 0 \\ -1 & 3 & -5 & | & 0 & 0 & 1 \end{bmatrix}$$

3. Step 2: Row Operations to Zero Out Elements Below the Leading 1
   To transform the matrix into row-echelon form, we now need to make the entries below the leading 1 in Row 1 (the first column) zero. The operation performed is:

   • For Row 2: $R_2 = R_2 - 3 \times R_1$
   • For Row 3: $R_3 = R_3 + R_1$

   Applying these operations yields:

   1 & -3 & 10 & | & 0 & 1 & 0 \\ 0 & 6 & -24 & | & 1 & -3 & 0 \\ 0 & 0 & 5 & | & 0 & 1 & 1 \end{bmatrix}$$

Summary of Row Operations

Each step involves simplifying the matrix to reach a form where we can isolate the identity matrix on the left side, which will lead us to the inverse on the right side.

Would you like further clarification on each row operation or steps for the next part of finding the inverse?

Here are some related questions that expand on this topic:

1. What are the next steps to completely reduce this matrix to the identity matrix?
2. How does the process of finding an inverse differ if the matrix is not invertible?
3. Why is row swapping necessary in certain cases during matrix inversion?
4. What other methods can be used to find the inverse of a matrix?
5. How can matrix inversion be applied to solve systems of linear equations?

Tip: When performing row operations, carefully keep track of each step to avoid errors, especially in the augmented part of the matrix where the identity matrix is being transformed.