Which of the following matrices can NOT be obtained from the matrix [-1, 2; 1, -1] by a single elementary row operation ?

Solution

Maths sir · 3 steps

See which row changed, then ask whether one single operation on that row can produce it.

1. Three allowed moves. A single elementary row operation does exactly one of these: swap two rows, multiply one row by a non-zero number, or add k times one row to another row. Here R₁ = (-1, 2) and R₂ = (1, -1).
2. Test the target row. In (c) and (d), R₁ is unchanged, so the operation acted on R₂ alone. Adding k times R₁ gives R₂ + kR₁ = (1-k, -1+2k), and scaling gives cR₂ = (c, -c).

   Option	What changed	Single operation?
   (a)	R₁ became (0, 1)	Yes: R₁ → R₁ + R₂
   (b)	Rows switched places	Yes: R₁ ↔ R₂
   (c)	R₂ became (-2, 7)	No: 1-k=-2 needs k=3, but -1+2k=7 needs k=4. Also (c, -c) cannot be (-2, 7)
   (d)	R₂ became (-1, 3)	Yes: R₂ → R₂ + 2R₁, since k=2 gives (-1, 3)

3. So. No single choice of k or scaling factor turns R₂ into (-2, 7). The matrix in (c) cannot be obtained in one step.

The idea to keep. For a new row of the form Rᵢ + kRⱼ, every entry must give the same k. If two entries need different values of k, then one operation is not enough.

Answer from the NTA final answer key. Solution by PrepWiser.