Which of the following matrices can NOT be obtained from the matrix [[-1, 2], [1, -1]] by a single e

Q: Wait, why can't we just use row multiplication to get to option (C)?

Great question! Row scaling ($R_i \to kR_i$) only allows you to multiply the *entire* row by a constant. If you multiply the second row of the original matrix by a constant, the ratio between the two elements in that row must remain the same. In option (C), the ratio changes, which is a tell-tale sign that a simple scaling operation won't work.

Q: How do I know if I should use row addition or row swapping?

Always start by checking the simplest operations first. Swapping is easy to spot—just look if the rows have traded places. If they haven't, check if one row is a linear combination of the other. If the target matrix looks like a 'shifted' version of the original, row addition is your best friend.

Q: Is it possible that multiple operations could lead to the same result?

Absolutely! But the question specifically asks which one *cannot* be obtained by a *single* operation. This means we are looking for the 'outlier' that requires two or more steps. If you find one that works in one step, you can safely move on to the next option.

Q: Why does $k$ have to be the same for both elements in a row?

Because an elementary row operation is a global transformation applied to the entire row. When you perform $R_2 \to R_2 + kR_1$, the scalar $k$ acts on the entire row vector. If you need different values of $k$ for different columns, you are essentially trying to perform two different operations at once, which is not allowed!

Q: Is there a shortcut to avoid testing every single option?

While testing is the most rigorous way, you can look at the determinants. Row swapping changes the sign of the determinant, and row addition leaves it unchanged. Row scaling multiplies the determinant by $k$. If the determinant of your target matrix doesn't relate to the original via these rules, you might find your answer faster!

Question

Which of the following matrices can NOT be obtained from the matrix $\begin{bmatrix} -1 & 2 \ 1 & -1 \end{bmatrix}$ by a single elementary row operation?

Options:

(A) $\begin{bmatrix} 0 & 1 \ 1 & -1 \end{bmatrix}$

(B) $\begin{bmatrix} 1 & -1 \ -1 & 2 \end{bmatrix}$

(C) $\begin{bmatrix} -1 & 2 \ -2 & 7 \end{bmatrix}$

(D) $\begin{bmatrix} -1 & 2 \ -1 & 3 \end{bmatrix}$

Sigma Percentile · Accepted Answer

### The Geometry of Transformation: Unlocking Matrix Operations

Welcome, fellow traveler on the path to JEE mastery! Today, we are not just solving a matrix problem; we are exploring the fundamental 'DNA' of linear algebra.

Imagine the matrix $A = \begin{bmatrix} -1 & 2 \ 1 & -1 \end{bmatrix}$ as a physical object in a two-dimensional space. Each row is a vector, and these vectors define the orientation of our coordinate system.

When we perform an elementary row operation, we are essentially 're-shaping' this space without destroying its core identity.

### The Three Pillars of Change

Before we dive into the options, let us remind ourselves of the tools in our toolkit. We have three, and only three, ways to manipulate these rows:

1. **Row Interchange ($R_i \leftrightarrow R_j$):** A simple swap. It is like looking at the world from a different perspective.

2. **Row Scaling ($R_i 	o kR_i$):** Stretching or shrinking a row by a non-zero factor $k$. It changes the magnitude but keeps the direction (or reverses it if $k < 0$).

3. **Row Addition ($R_i 	o R_i + kR_j$):** This is the most powerful tool. It is the 'shear' transformation. We are adding a piece of one row to another, effectively tilting our perspective.

Our goal is to identify which of the given matrices cannot be reached by applying exactly one of these operations to our original matrix $A$.

### The Investigation

Let us test our options one by one. Think of this as a detective story where we check the alibi of each matrix.

For **Option (A)**, we look at $\begin{bmatrix} 0 & 1 \ 1 & -1 \end{bmatrix}$. If we apply the operation $R_1 	o R_1 + R_2$, we get:

$$
\begin{bmatrix} -1+1 & 2+(-1) \ 1 & -1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \ 1 & -1 \end{bmatrix}
$$

It works! The matrix is reachable. The 'alibi' holds.

For **Option (B)**, we see $\begin{bmatrix} 1 & -1 \ -1 & 2 \end{bmatrix}$. This looks suspiciously like the rows have swapped places.

If we perform $R_1 \leftrightarrow R_2$, we get exactly this result. This is a perfectly valid single operation.

For **Option (D)**, we test $\begin{bmatrix} -1 & 2 \ -1 & 3 \end{bmatrix}$. Let us try the operation $R_2 	o R_2 + 2R_1$:

$$
\begin{bmatrix} -1 & 2 \ 1+2(-1) & -1+2(2) \end{bmatrix} = \begin{bmatrix} -1 & 2 \ -1 & 3 \end{bmatrix}
$$

Again, a perfect match. The operation is valid.

### The Impossible Case

Now, we arrive at **Option (C)**: $\begin{bmatrix} -1 & 2 \ -2 & 7 \end{bmatrix}$. We want to see if we can reach this via $R_2 	o R_2 + kR_1$.

Let us set up the algebra for the elements of the second row:

For the first column: $1 + k(-1) = -2$. Solving this gives us $k = 3$.

Now, we must check if this same $k$ works for the second column: $-1 + k(2) = 7$. Substituting $k=3$, we get $-1 + 3(2) = 5$.

But wait! $5 
eq 7$. Because the $k$ required for the first column does not satisfy the second column, there is no single elementary row operation that can transform the original matrix into this one. It is mathematically impossible in one step.

### Final Reflections

Why does this matter? Because in the world of linear algebra, these operations are the building blocks of Gaussian elimination.

Understanding what you *cannot* do is just as important as knowing what you *can* do. You have successfully navigated the constraints of row operations.

Keep this intuition sharp—it is the difference between memorizing steps and truly understanding the elegance of the math. You are doing great!

Animated Solution for Mathematics - Matrices and Determinants: Which of the following matrices can NOT be obtained from the matrix $[- 1 1 2 - 1]$ by a single elementary row operation?

Select Answer:

Visualized Solution

Identify the Original Matrix $A$

Recall Elementary Row Operations

Testing Option (A)

Testing Option (B)

Testing Option (D)

Testing Option (C) - The Impossible Case

Final Conclusion

The Sigma Insight: Algebraic Operations on Matrices

The Geometry of Transformation

The Three Pillars of Change

The Investigation

The Impossible Case

Final Reflections

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