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 $B$.

   \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.