1. Key Concepts and Formulas

• Elementary Row Operations (EROs): These are fundamental transformations applied to the rows of a matrix. There are three types:
  1. Row Swap: Interchanging two rows ($R_i \leftrightarrow R_j$). This operation changes the sign of the determinant.
  2. Row Scaling: Multiplying a row by a non-zero scalar ($R_i \to kR_i$, where $k \neq 0$). This operation multiplies the determinant by $k$.
  3. Row Addition: Adding a scalar multiple of one row to another row ($R_i \to R_i + kR_j$, where $i \neq j$). This operation leaves the determinant unchanged.
• Determinant Property: The determinant of a matrix can be used as a quick check for elementary row operations. If a matrix $B$ is obtained from matrix $A$ by a single ERO, their determinants will be related as described above. If the determinant relationship does not hold, then $B$ cannot be obtained from $A$ by that specific type of ERO.

1. Step-by-Step Solution

Let the given matrix be $A = \left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$. Its determinant is $\det(A) = (-1)(-1) - (2)(1) = 1 - 2 = -1$. We will check each option to see if it can be obtained from $A$ by a single elementary row operation.

Step 1: Analyze Option (A) The matrix is $M_A = \left[\begin{array}{cc}0 & 1 \\ 1 & -1\end{array}\right]$. Its determinant is $\det(M_A) = (0)(-1) - (1)(1) = 0 - 1 = -1$.

• Check Row Swap ($R_1 \leftrightarrow R_2$): Swapping rows of $A$ gives $\left[\begin{array}{cc}1 & -1 \\ -1 & 2\end{array}\right]$, which is not $M_A$.
• Check Row Scaling ($R_i \to kR_i$):
  • If $R_1 \to kR_1$: We need $k[-1 \quad 2] = [0 \quad 1]$. This implies $-k=0 \implies k=0$, which is not allowed for row scaling.
  • If $R_2 \to kR_2$: We need $k[1 \quad -1] = [1 \quad -1]$. This implies $k=1$. If $R_2 \to R_2$ (i.e., $k=1$), $R_1$ must remain unchanged. The resulting matrix would be $A$ itself, not $M_A$.
• Check Row Addition ($R_i \to R_i + kR_j$):
  • If $R_1 \to R_1 + kR_2$: The first row of $A$ is $[-1 \quad 2]$ and the second row is $[1 \quad -1]$. We want $[-1 \quad 2] + k[1 \quad -1] = [0 \quad 1]$. This gives two equations:
    1. $-1 + k = 0 \implies k=1$.
    2. $2 - k = 1 \implies k=1$. Since we found a consistent value $k=1$, the operation $R_1 \to R_1 + R_2$ transforms $A$ into $M_A$: $\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right] \xrightarrow{R_1 \to R_1 + R_2} \left[\begin{array}{cc}-1+1 & 2+(-1) \\ 1 & -1\end{array}\right] = \left[\begin{array}{cc}0 & 1 \\ 1 & -1\end{array}\right]$
  • If $R_2 \to R_2 + kR_1$: The first row of $M_A$ is $[0 \quad 1]$, which is not the first row of $A$. So this operation would also need $R_1$ to be changed, which is not allowed for a single operation affecting only $R_2$. If $R_1$ is unchanged, then $k$ must be 0, which would leave $R_2$ unchanged, resulting in $A$ itself.
• Conclusion for (A): The matrix $M_A$ can be obtained from $A$ by the operation $R_1 \to R_1 + R_2$.

Step 2: Analyze Option (B) The matrix is $M_B = \left[\begin{array}{cc}1 & -1 \\ -1 & 2\end{array}\right]$. Its determinant is $\det(M_B) = (1)(2) - (-1)(-1) = 2 - 1 = 1$.

• Check Row Swap ($R_1 \leftrightarrow R_2$): Swapping the rows of $A$: $\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right] \xrightarrow{R_1 \leftrightarrow R_2} \left[\begin{array}{cc}1 & -1 \\ -1 & 2\end{array}\right]$
• This is exactly $M_B$. Also, $\det(M_B) = -\det(A) = -(-1) = 1$, which is consistent.
• Conclusion for (B): The matrix $M_B$ can be obtained from $A$ by the operation $R_1 \leftrightarrow R_2$.

Step 3: Analyze Option (C) The matrix is $M_C = \left[\begin{array}{rr}-1 & 2 \\ -2 & 7\end{array}\right]$. Its determinant is $\det(M_C) = (-1)(7) - (2)(-2) = -7 + 4 = -3$.

• Compare rows of $A$ and $M_C$: The first row of $M_C$ is $[-1 \quad 2]$, which is identical to $R_1$ of $A$. This means any elementary row operation must have affected only $R_2$.
• Check Row Swap ($R_1 \leftrightarrow R_2$): Not possible, as $R_1$ is unchanged.
• Check Row Scaling ($R_2 \to kR_2$): We need $k[1 \quad -1] = [-2 \quad 7]$. This implies $k = -2$ (from the first component) and $-k = 7 \implies k = -7$ (from the second component). Since $k$ must be a single consistent value, this operation is not possible.
• Check Row Addition ($R_2 \to R_2 + kR_1$): We need $[1 \quad -1] + k[-1 \quad 2] = [-2 \quad 7]$. This gives two equations:
  1. $1 - k = -2 \implies k=3$.
  2. $-1 + 2k = 7 \implies 2k=8 \implies k=4$. Since $k$ must be a single consistent value, this operation is not possible.
• Check with Determinant: $\det(M_C) = -3$.
  • If it was a row swap, $\det(M_C)$ would be $1$. (Not possible)
  • If it was $R_1 \to kR_1$, then $\det(M_C) = k \det(A) = -k$. So $-3 = -k \implies k=3$. But $R_1$ is unchanged in $M_C$.
  • If it was $R_2 \to kR_2$, then $\det(M_C) = k \det(A) = -k$. So $-3 = -k \implies k=3$. We already showed this is inconsistent by comparing row elements.
  • If it was $R_i \to R_i + kR_j$, then $\det(M_C)$ would be $\det(A) = -1$. But $\det(M_C) = -3$. (Not possible)
• Conclusion for (C): The matrix $M_C$ cannot be obtained from $A$ by a single elementary row operation.

Step 4: Analyze Option (D) The matrix is $M_D = \left[\begin{array}{ll}-1 & 2 \\ -1 & 3\end{array}\right]$. Its determinant is $\det(M_D) = (-1)(3) - (2)(-1) = -3 + 2 = -1$.

• Compare rows of $A$ and $M_D$: The first row of $M_D$ is $[-1 \quad 2]$, which is identical to $R_1$ of $A$. This means any elementary row operation must have affected only $R_2$.
• Check Row Swap ($R_1 \leftrightarrow R_2$): Not possible, as $R_1$ is unchanged.
• Check Row Scaling ($R_2 \to kR_2$): We need $k[1 \quad -1] = [-1 \quad 3]$. This implies $k = -1$ (from the first component) and $-k = 3 \implies k = -3$ (from the second component). Since $k$ must be a single consistent value, this operation is not possible.
• Check Row Addition ($R_2 \to R_2 + kR_1$): We need $[1 \quad -1] + k[-1 \quad 2] = [-1 \quad 3]$. This gives two equations:
  1. $1 - k = -1 \implies k=2$.
  2. $-1 + 2k = 3 \implies 2k=4 \implies k=2$. Since we found a consistent value $k=2$, the operation $R_2 \to R_2 + 2R_1$ transforms $A$ into $M_D$: $\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right] \xrightarrow{R_2 \to R_2 + 2R_1} \left[\begin{array}{cc}-1 & 2 \\ 1+2(-1) & -1+2(2)\end{array}\right] = \left[\begin{array}{cc}-1 & 2 \\ -1 & 3\end{array}\right]$
• Conclusion for (D): The matrix $M_D$ can be obtained from $A$ by the operation $R_2 \to R_2 + 2R_1$.

Based on our thorough analysis, options (A), (B), and (D) can be obtained from the given matrix by a single elementary row operation, while option (C) cannot.

1. Common Mistakes & Tips

• Forgetting $k \neq 0$ for Row Scaling: Always ensure the scalar $k$ used in $R_i \to kR_i$ is non-zero. If $k=0$, it's not a valid elementary row operation.
• Inconsistent $k$ values: When solving for $k$ in row addition or scaling, ensure the value of $k$ is consistent across all components of the row. If different components yield different $k$ values, that operation is not possible.
• Determinant as a quick check: Use determinant properties to quickly rule out possibilities. For example, if a row addition operation (which preserves determinant) is suspected, but the determinants are different, that operation is impossible.
• Systematic Checking: Go through each of the three types of elementary row operations for each row to ensure no possibility is missed.

1. Summary

We systematically checked each given option to determine if it could be obtained from the initial matrix by a single elementary row operation. We considered row swaps, row scaling, and row addition operations. We found that options (A), (B), and (D) can all be obtained through a single elementary row operation. Specifically, (A) is obtained by $R_1 \to R_1 + R_2$, (B) by $R_1 \leftrightarrow R_2$, and (D) by $R_2 \to R_2 + 2R_1$. However, for option (C), no consistent scalar $k$ could be found for either row scaling or row addition operations, nor did a row swap apply, making it the matrix that cannot be obtained.

1. Final Answer

The final answer is \boxed{A}