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The given matrix is an elementary matrix, and we need to identify the elementary row operation associated with it. 

### Matrix:
\[
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & \frac{1}{3} & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}
\]

This matrix differs from the identity matrix in the second row, where the element in the \( (2,4) \)-position is \( \frac{1}{3} \). This suggests that a multiple of row 4 has been added to row 2. Specifically, this corresponds to the operation:

\[
R_2 \to R_2 + \frac{1}{3} R_4
\]

Therefore, the correct answer is **E: \( R_2 + \frac{1}{3} R_4 \)**.

Would you like more details on how elementary row operations work, or do you have any further questions?

### Here are 5 questions for further exploration:
1. What are the types of elementary row operations and how do they affect matrices?
2. How can elementary matrices be used to find the inverse of a matrix?
3. Can you explain how Gaussian elimination uses elementary row operations?
4. What is the relationship between row operations and the determinant of a matrix?
5. How does performing row operations help solve systems of linear equations?

**Tip:** Elementary matrices are useful tools because they represent the exact row operation you performed on an identity matrix, and they are invertible!

What is the elementary row operation associated with the following elementary matrix?

Discover the elementary row operation for a given matrix transformation. This problem focuses on the operation R2 → R2 + (1/3)R4, and involves key concepts from linear algebra, such as elementary matrices and matrix transformations.

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