The task requires performing a series of row operations on matrices, and we need to apply the operations in the given order.

Row operations provided:

1. $-2R_1 + R_2 \rightarrow R_2$
2. $-R_1 + R_3 \rightarrow R_3$
3. $-\frac{1}{3} R_2 \rightarrow R_2$
4. $R_2 + R_3 \rightarrow R_3$
5. $\frac{1}{3} R_3 \rightarrow R_3$

To proceed:

1. We start with a matrix (denote it as $A$).
2. Apply each operation one by one to change the rows of the matrix.
3. After each operation, we get an updated matrix.

Let's walk through the operations step by step.

Initial Matrix

Let the initial matrix be denoted as: