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To solve this, we need to perform matrix multiplication between the elementary matrix \( E \) and matrix \( A \), as requested:

Given:
\[
E = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 3 & 6 & 1 \\ 7 & 9 & 2 \\ 4 & 8 & -1 \end{bmatrix}
\]

### Step-by-Step Solution

1. **Multiply** each row of \( E \) with each column of \( A \).
   
   The resulting matrix \( EA \) will have the same dimensions as \( A \) (3x3). Each element in \( EA \) is found by taking the dot product of the corresponding row in \( E \) with each column in \( A \).

   - **First row of \( EA \)**:
     \[
     (1 \times 3) + (0 \times 7) + (0 \times 4) = 3
     \]
     \[
     (1 \times 6) + (0 \times 9) + (0 \times 8) = 6
     \]
     \[
     (1 \times 1) + (0 \times 2) + (0 \times -1) = 1
     \]
     So, the first row of \( EA \) is \([3, 6, 1]\).

   - **Second row of \( EA \)**:
     \[
     (0 \times 3) + (4 \times 7) + (0 \times 4) = 28
     \]
     \[
     (0 \times 6) + (4 \times 9) + (0 \times 8) = 36
     \]
     \[
     (0 \times 1) + (4 \times 2) + (0 \times -1) = 8
     \]
     So, the second row of \( EA \) is \([28, 36, 8]\).

   - **Third row of \( EA \)**:
     \[
     (0 \times 3) + (0 \times 7) + (1 \times 4) = 4
     \]
     \[
     (0 \times 6) + (0 \times 9) + (1 \times 8) = 8
     \]
     \[
     (0 \times 1) + (0 \times 2) + (1 \times -1) = -1
     \]
     So, the third row of \( EA \) is \([4, 8, -1]\).

2. **Combine these results** to form the final matrix \( EA \):
   \[
   EA = \begin{bmatrix} 3 & 6 & 1 \\ 28 & 36 & 8 \\ 4 & 8 & -1 \end{bmatrix}
   \]

This is the result for \( EA \).

Would you like a more detailed explanation, or do you have any questions?

### Further Questions
1. What is an elementary matrix, and how does it affect other matrices?
2. How does matrix multiplication work in general, and what are its properties?
3. Can elementary matrices be used for finding inverses?
4. Why does multiplying a matrix by an elementary matrix correspond to row operations?
5. How would the result change if a different elementary matrix were used?

### Tip
In matrix multiplication, make sure the dimensions align: the number of columns in the first matrix must match the number of rows in the second matrix.

Use the elementary matrix E to find EA using an elementary row operation.

Step-by-step solution for using an elementary matrix to find the product EA. This problem explores key linear algebra concepts like matrix multiplication and elementary row operations, ideal for undergraduate students studying matrix theory.

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