Which one of the following matrices is an elementary matrix?
\(\left[ {\begin{array}{c} 1&5&0\\ 0&1&0\\ 0&0&1 \end{array}} \right]\)
Understanding Elementary Matrices
An elementary matrix is a matrix that is obtained by performing exactly one elementary row operation on an identity matrix. The elementary row operations are:
- Swapping two rows.
- Multiplying a row by a non-zero scalar.
- Adding a multiple of one row to another row.
We need to check each given matrix to see if it can be obtained by applying one of these operations to the 3x3 identity matrix, which is:
\(I = \left[ {\begin{array}{c} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right]\)
Analyzing Each Option
Option 1: \( \left[ {\begin{array}{c} 1&0&0\\ 0&0&0\\ 0&0&1 \end{array}} \right] \)
This matrix has a row of zeros (the second row). A row of zeros can only be obtained from the identity matrix by multiplying a row by zero. However, multiplying a row by zero is not considered an elementary row operation because the scalar must be non-zero. Therefore, this matrix is not an elementary matrix.
Option 2: \( \left[ {\begin{array}{c} 1&5&0\\ 0&1&0\\ 0&0&1 \end{array}} \right] \)
Let's start with the identity matrix \(I\). Consider applying the operation of adding 5 times the second row (\(R_2\)) to the first row (\(R_1\)), denoted as \(R_1 \leftarrow R_1 + 5R_2\).
Applying this operation to the identity matrix \(I\):
\( \left[ {\begin{array}{c} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right] \xrightarrow{R_1 \leftarrow R_1 + 5R_2} \left[ {\begin{array}{c} 1 + 5(0) & 0 + 5(1) & 0 + 5(0)\\ 0&1&0\\ 0&0&1 \end{array}} \right] = \left[ {\begin{array}{c} 1&5&0\\ 0&1&0\\ 0&0&1 \end{array}} \right] \)
This result matches the matrix in Option 2. Since this matrix was obtained by performing exactly one elementary row operation on the identity matrix, it is an elementary matrix.
Option 3: \( \left[ {\begin{array}{c} 0&2&0\\ 1&0&0\\ 0&0&1 \end{array}} \right] \)
Let's examine if this can be obtained from the identity matrix by a single operation.
- Swapping R1 and R2 gives \( \left[ {\begin{array}{c} 0&1&0\\ 1&0&0\\ 0&0&1 \end{array}} \right] \), which is not the given matrix.
- Multiplying a row by a scalar wouldn't produce the 0 in the (1,1) position unless the original entry was 0, which is not the case in the identity matrix.
- Adding a multiple of one row to another wouldn't create the combination of 0 in (1,1) and 2 in (1,2) from the identity matrix's first row (1,0,0).
This matrix requires swapping rows 1 and 2 and then multiplying the new row 1 by 2. This involves two operations, so it is not an elementary matrix.
Option 4: \( \left[ {\begin{array}{c} 1&0&0\\ 0&1&0\\ 0&5&2 \end{array}} \right] \)
Let's examine this matrix. Compared to the identity matrix, the entry at (3,3) is 2 instead of 1, and the entry at (3,2) is 5 instead of 0.
- The '2' in the (3,3) position suggests the third row was multiplied by 2 (\(R_3 \leftarrow 2R_3\)). This operation on the identity gives \( \left[ {\begin{array}{c} 1&0&0\\ 0&1&0\\ 0&0&2 \end{array}} \right] \).
- The '5' in the (3,2) position suggests a multiple of row 2 was added to row 3 (\(R_3 \leftarrow R_3 + cR_2\)). To get 5, we would need \(c=5\) if starting from identity. This operation \(R_3 \leftarrow R_3 + 5R_2\) on the identity gives \( \left[ {\begin{array}{c} 1&0&0\\ 0&1&0\\ 0&5&1 \end{array}} \right] \).
Neither a single scaling of \(R_3\) nor a single addition of a multiple of \(R_2\) to \(R_3\) produces the matrix in Option 4. It requires a combination of operations (scaling and adding). Therefore, this matrix is not an elementary matrix.
Conclusion
Based on the analysis, only the matrix in Option 2 can be obtained from the identity matrix by a single elementary row operation (adding 5 times the second row to the first row). Thus, it is the only elementary matrix among the given options.
Revision Table: Identifying Elementary Matrices
| Matrix | Relation to Identity Matrix | Elementary Matrix? | Reason |
|---|---|---|---|
| \( \left[ {\begin{array}{c} 1&0&0\\ 0&0&0\\ 0&0&1 \end{array}} \right] \) | Row 2 is \(0 \times\) Identity \(R_2\) | No | Scalar must be non-zero for scaling operation. |
| \( \left[ {\begin{array}{c} 1&5&0\\ 0&1&0\\ 0&0&1 \end{array}} \right] \) | \(R_1 \leftarrow R_1 + 5R_2\) on Identity | Yes | Single elementary row operation of type 3. |
| \( \left[ {\begin{array}{c} 0&2&0\\ 1&0&0\\ 0&0&1 \end{array}} \right] \) | Requires swapping \(R_1, R_2\) and scaling \(R_1\) | No | Requires two operations. |
| \( \left[ {\begin{array}{c} 1&0&0\\ 0&1&0\\ 0&5&2 \end{array}} \right] \) | Requires scaling \(R_3\) and adding \(5R_2\) to \(R_3\) | No | Requires two operations. |
Additional Information on Elementary Matrices
Elementary matrices are fundamental in linear algebra because multiplying a matrix \(A\) by an elementary matrix \(E\) on the left (\(EA\)) performs the same elementary row operation on \(A\) that was performed on the identity matrix to obtain \(E\). Similarly, multiplying by an elementary matrix on the right performs a column operation.
Properties of elementary matrices:
- Every elementary matrix is invertible. The inverse of an elementary matrix is also an elementary matrix.
- Any invertible matrix can be written as a product of elementary matrices. This is a key result used in understanding matrix invertibility and Gaussian elimination.
They provide a matrix representation for elementary row operations, linking the abstract operations to concrete matrix multiplication.
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