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We are given an elementary matrix, and the task is to identify the elementary row operation that corresponds to this matrix.

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

### Analyzing the Matrix:
- This matrix is almost the identity matrix, except for one nonzero off-diagonal entry in the second row, fourth column (\( \frac{1}{3} \)).
- The entry in the second row and fourth column indicates that we have added \(\frac{1}{3}\) of the fourth row (\(R_4\)) to the second row (\(R_2\)).

### Conclusion:
This operation corresponds to:
\[
R_2 \rightarrow R_2 + \frac{1}{3}R_4
\]
Thus, the correct answer is **E**: \( R_2 + \frac{1}{3}R_4 \).

Would you like further clarification on this or details on elementary matrices?

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Here are 5 related questions you might find useful:
1. What are the three types of elementary row operations?
2. How do elementary matrices relate to solving linear systems?
3. How do you find the inverse of an elementary matrix?
4. What is the significance of elementary matrices in Gaussian elimination?
5. How do elementary row operations affect the determinant of a matrix?

**Tip**: Every elementary matrix represents a single elementary row operation, and applying that matrix to another matrix mimics performing the row operation.

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

Discover the elementary row operation represented by a given elementary matrix. Learn about elementary matrices and row operations, specifically the operation R2 -> R2 + (1/3)R4. Ideal for students studying linear algebra in Grades 10-12.

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