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

\[
B = E_2 E_1 A
\]

where:

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

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

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

### Solution Explanation:

- \( E_1 \) is a matrix that scales the first row of matrix \( A \) by 2. This operation transforms the first row of \( A \), which is \([7 \,\, 1]\), to \([14 \,\, 2]\). This gives us an intermediate matrix after multiplication by \( E_1 \):

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

- \( E_2 \) is an elementary row operation that adds the first row to the second row. When applied to the intermediate matrix \( E_1 A \), it transforms it into matrix \( B \):

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

Thus, the elementary matrices \( E_1 \) and \( E_2 \) produce matrix \( B \) when multiplied by matrix \( A \) as required.

Would you like further details on this process, or do you have any questions?

### Related Questions:
1. How are elementary matrices related to row operations?
2. Can an elementary matrix be a non-invertible matrix?
3. How do we identify the correct elementary matrix for a given row operation?
4. What is the geometric interpretation of applying elementary matrices?
5. How can we extend this method to larger matrices?

### Tip:
Elementary matrices are key tools in performing row operations, which are essential in algorithms like Gaussian elimination to solve systems of linear equations or find matrix inverses.

Find elementary matrices \( E_1 \) and \( E_2 \) so that \( B = E_2 E_1 A \), where \( A = \begin{bmatrix} 7 & 1 \\ 0 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 14 & 2 \\ 7 & 1 \end{bmatrix} \).

Discover how to find elementary matrices \( E_1 \) and \( E_2 \) to achieve the transformation \( B = E_2 E_1 A \). This problem illustrates key concepts in linear algebra, including matrix multiplication and row operations, perfect for undergraduate students studying matrix theory.

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