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The exercise asks:

"**Exercise 2.5.4**: If \( E \) is elementary, show that \( A \) and \( EA \) differ in at most two rows."

### Explanation:
An **elementary matrix** is a matrix derived from the identity matrix by performing a single elementary row operation (row swaps, scalar multiplication, or adding multiples of rows). When multiplying an elementary matrix \( E \) by another matrix \( A \), the product \( EA \) represents the effect of applying that elementary row operation to matrix \( A \).

To show that \( A \) and \( EA \) differ in at most two rows:

1. If \( E \) corresponds to a row swap operation, then only two rows of \( A \) are swapped in \( EA \), so \( A \) and \( EA \) will differ in exactly two rows.
  
2. If \( E \) corresponds to multiplying a row of the identity by a scalar, then only one row in \( A \) will be multiplied by that scalar, meaning \( A \) and \( EA \) will differ in exactly one row.

3. If \( E \) corresponds to adding a multiple of one row to another, this operation will affect only one row of \( A \), so again, \( A \) and \( EA \) will differ in one row.

In all these cases, \( A \) and \( EA \) will differ in at most two rows. Therefore, the statement is true for all elementary matrices \( E \).

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Would you like more details on any step? Here are five related questions to deepen understanding:

1. What are the three types of elementary row operations?
2. How can an elementary matrix be constructed from the identity matrix?
3. Why do elementary matrices always represent invertible operations?
4. How does matrix multiplication relate to applying row operations?
5. Can elementary matrices be used to compute the inverse of a matrix?

**Tip**: To understand the effect of elementary matrices, practice applying different elementary row operations to a small matrix, and observe how the rows change.

Exercise 2.5.4: If E is elementary, show that A and EA differ in at most two rows.

Learn how to show that when an elementary matrix E multiplies matrix A, the resulting matrix EA differs from A in at most two rows. This problem involves key concepts from linear algebra, including elementary matrices and matrix operations. Suitable for college-level linear algebra students.

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