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.