Exercise 3.48. Are A =


0 1 0
0 0 1
1 0 0

 and B =
1 0
0 0

elementary matrices

Let's determine whether the given matrices \( A \) and \( B \) are elementary matrices.

### Definition of Elementary Matrices:
An **elementary matrix** is a matrix that can be obtained by performing a single elementary row operation on an identity matrix. There are three types of elementary row operations:
1. **Row swapping**: Interchanging two rows.
2. **Row scaling**: Multiplying a row by a non-zero scalar.
3. **Row addition**: Adding a multiple of one row to another row.

### Matrix A:
Matrix \( A \) is given as:

\[
A = \begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{pmatrix}
\]

- Compare \( A \) with the 3x3 identity matrix \( I_3 \):

\[
I_3 = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]

Matrix \( A \) can be obtained by performing the following **row swapping operations** on \( I_3 \):
- Swap row 1 with row 3.
- Swap row 2 with row 1.

Since \( A \) can be obtained by performing row swaps on the identity matrix, **matrix \( A \) is an elementary matrix**.

### Matrix B:
Matrix \( B \) is given as:

\[
B = \begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
\]

- Compare \( B \) with the 2x2 identity matrix \( I_2 \):

\[
I_2 = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\]

Matrix \( B \) cannot be obtained by performing any of the three types of elementary row operations (row swapping, scaling, or addition) on the identity matrix. Therefore, **matrix \( B \) is not an elementary matrix**.

### Conclusion:
- **Matrix \( A \) is an elementary matrix**.
- **Matrix \( B \) is not an elementary matrix**.

Would you like further explanation or details on elementary matrices?

Here are 5 related questions to expand on this topic:
1. What are the three types of elementary row operations?
2. How does multiplying a row by a scalar affect whether a matrix is elementary?
3. Can a non-square matrix be an elementary matrix?
4. How do elementary matrices relate to row-reduction techniques?
5. What is the inverse of an elementary matrix, and is it also an elementary matrix?

**Tip**: Elementary matrices are crucial in understanding matrix inverses and solving systems of linear equations. They represent the steps taken during row-reduction!

Exercise 3.48. Are A = [ [0, 1, 0], [0, 0, 1], [1, 0, 0] ] and B = [ [1, 0], [0, 0] ] elementary matrices?

Analyze whether matrices A and B are elementary matrices through row operations. Explore core concepts in linear algebra such as elementary row operations and identity matrices in this problem, designed for undergraduate students.

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