Which one of the following matrices is an elementary matrix ?

Options

A. `[{:(1,0,0),(0,0,0),(0,0,1):}]`
B. `[{:(1,5,0),(0,1,0),(0,0,1):}]`
C. `[{:(0,2,0),(1,0,0),(0,0,1):}]`
D. `[{:(1,0,0),(0,1,0),(0,5,2):}]`

Explanation

To determine which matrix is an elementary matrix, we first need to understand what an elementary matrix is. An elementary matrix is derived from the identity matrix by performing a single elementary row operation. The characteristics of an elementary matrix include:

1. The diagonal elements are all 1.
2. The off-diagonal elements can be either 0 or some other value depending on the operation performed.

Now, let's analyze the given matrices step by step to identify the elementary matrix.

### Step 1: Identify the Matrices
Assume we have the following matrices to analyze:
- Matrix A:
\[ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
- Matrix B:
\[ \begin{pmatrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
- Matrix C:
\[ \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \]
- Matrix D:
\[ \begin{pmatrix} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 2: Check Each Matrix
- **Matrix A**:
- Diagonal elements: 1, 1, 1 (all are 1)
- Off-diagonal elements: 0
- Conclusion: This is an elementary matrix (specifically, the identity matrix).

- **Matrix B**:
- Diagonal elements: 1, 1, 1 (all are 1)
- Off-diagonal elements: 2 (not zero but acceptable in an elementary matrix)
- Conclusion: This is also an elementary matrix (it represents a row operation).

- **Matrix C**:
- Diagonal elements: 0, 0, 0 (not all are 1)
- Conclusion: This is not an elementary matrix.

- **Matrix D**:
- Diagonal elements: 2, 0, 1 (not all are 1)
- Conclusion: This is not an elementary matrix.

### Step 3: Final Conclusion
From the analysis, both Matrix A and Matrix B qualify as elementary matrices. However, if we are to choose one, Matrix B is the one that represents a specific row operation.

### Final Answer:
**Matrix B is an elementary matrix.**

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