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The problem asks to find elementary matrices \( E_1 \) and \( E_2 \) such that:

\[
B = E_2 E_1 A
\]

where

\[
A = \begin{bmatrix} 7 & 1 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 14 & 2 \\ 7 & 1 \end{bmatrix}
\]

### Strategy
Elementary matrices represent row operations. To find \( E_1 \) and \( E_2 \), we need to identify the sequence of row operations that transforms matrix \( A \) into matrix \( B \).

Let's decompose the problem step by step:

1. **From A to B**:
   We need to perform operations that take \( A \) and turn it into \( B \).

   \[
   A = \begin{bmatrix} 7 & 1 \\ 0 & 0 \end{bmatrix}
   \quad \longrightarrow \quad
   B = \begin{bmatrix} 14 & 2 \\ 7 & 1 \end{bmatrix}
   \]

2. **Step 1: Doubling the first row**: 
   The first row of \( A \) needs to be doubled to turn \( 7 \) into \( 14 \) and \( 1 \) into \( 2 \).

   This can be done by multiplying the first row of \( A \) by 2, which corresponds to applying an elementary matrix \( E_1 \):

   \[
   E_1 = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}
   \]
   Applying \( E_1 \) to \( A \):

   \[
   E_1 A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 7 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 14 & 2 \\ 0 & 0 \end{bmatrix}
   \]

3. **Step 2: Adding the first row to the second row**:
   To get the second row of \( B \) (which is \( [7, 1] \)), we can add the first row to the second row. This corresponds to applying another elementary matrix \( E_2 \):

   \[
   E_2 = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}
   \]
   Applying \( E_2 \) to \( E_1 A \):

   \[
   E_2 E_1 A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 14 & 2 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 14 & 2 \\ 7 & 1 \end{bmatrix}
   \]

Thus, the elementary matrices \( E_1 \) and \( E_2 \) are:

\[
E_1 = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}, \quad E_2 = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}
\]

Would you like further explanation or details on any of the steps?

---

Here are 5 follow-up questions to consider:

1. What are elementary matrices, and how do they relate to row operations?
2. Can you reverse the process and find \( E_1^{-1} \) and \( E_2^{-1} \)?
3. How would the process change if matrix \( A \) had different values?
4. What types of row operations correspond to each elementary matrix?
5. How do elementary matrices help in solving systems of linear equations?

**Tip**: When working with elementary matrices, keep in mind that each matrix represents a single, simple row operation like row swapping, scaling, or adding one row to another.

Find elementary matrices E1 and E2 so that B = E2 E1 A where A = [7 1; 0 0], B = [14 2; 7 1].

This problem demonstrates how to find elementary matrices E1 and E2 that transform matrix A = [7 1; 0 0] to matrix B = [14 2; 7 1]. Using elementary matrix operations, we perform row operations to obtain the final result.

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