`[(1,-1),(2,3)]=[(1,0),(0,1)] A,R_(2) rarr R_(2)-2R_(1)`gives

Options

A. `[(1,-1),(0,1)]=[(1,0),(-2,1)]A`
B. `[(1,-1),(0,5)]=[(1,0),(-2,1)]A`
C. `[(1,-1),(0,5)]=[(1,0),(2,1)]A`
D. `[(1,-1),(0,5)]=[(1,0),(-2,0)]A`

Answer

To solve the problem given, we need to perform the row operation specified on the matrix. The matrix we have is:

\[ A = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix} \]

We are instructed to perform the row operation \( R_2 \rightarrow R_2 - 2R_1 \).

### Step-by-Step Solution:

1. **Identify the rows**:
- The first row \( R_1 \) is \( (1, -1) \).
- The second row \( R_2 \) is \( (2, 3) \).

2. **Calculate \( 2R_1 \)**:
- Multiply each element of \( R_1 \) by 2:
\[ 2R_1 = 2 \times (1, -1) = (2, -2) \]

3. **Perform the row operation**:
- Now, we will subtract \( 2R_1 \) from \( R_2 \):
\[ R_2 = R_2 - 2R_1 = (2, 3) - (2, -2) \]
- Subtract the corresponding elements:
\[ R_2 = (2 - 2, 3 - (-2)) = (0, 3 + 2) = (0, 5) \]

4. **Write the new matrix**:
- After performing the row operation, the new matrix becomes:
\[ A' = \begin{pmatrix} 1 & -1 \\ 0 & 5 \end{pmatrix} \]

### Final Answer:
The resulting matrix after the row operation \( R_2 \rightarrow R_2 - 2R_1 \) is:

\[ \begin{pmatrix} 1 & -1 \\ 0 & 5 \end{pmatrix} \]