We are given two matrices $A$ and $B$ and asked to find an elementary matrix $E$ such that $B = EA$.

The matrices are: $A = \begin{pmatrix} 5 & 6 & 5 \\ 6 & 4 & 4 \\ 3 & 7 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 6 & 4 & 4 \\ 5 & 6 & 5 \\ 3 & 7 & 5 \end{pmatrix}$

To find $E$, we need to examine how we can transform $A$ into $B$ by applying elementary row operations. By comparing the two matrices, it looks like rows 1 and 2 of $A$ have been swapped to obtain $B$, while row 3 remains unchanged.

Thus, $E$ is the elementary matrix corresponding to swapping rows 1 and 2. The elementary matrix for this operation is:

$E = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

This matrix swaps the first and second rows when multiplied by any 3x3 matrix from the left.

Would you like further details on how to verify this solution or more explanations on elementary matrices?

Here are 5 follow-up questions related to this problem:

1. What are the different types of elementary matrices, and how do they correspond to row operations?
2. How can you compute $B$ by directly multiplying $E$ and $A$?
3. Can an elementary matrix be undone? If so, how would you find its inverse?
4. What role do elementary matrices play in solving systems of linear equations?
5. How can we use elementary matrices to perform LU decomposition of a matrix?

Tip: Elementary matrices are always invertible, and their inverse corresponds to reversing the row operation they represent.