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:

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!