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Part CCSIR NET June 2025a-vector-killed-by-a-transpose-confines-the-whole-image-to-a-hyperplane

A vector killed by a transpose confines the whole image to a hyperplane

Let M2(R)M_{2}(\mathbb{R}) denote the space of real 2 × 2 matrices. Let S be the vector subspace of M2(R)M_{2}(\mathbb{R}) comprising of all symmetric matrices. Let F:M2(R)→SF : M_{2}(\mathbb{R}) \to S be the map defined by F(X) = XXᵀ. Let DFA:M2(R)→S_A : M_{2}(\mathbb{R}) \to S be the derivative of F at A∈M2(R)A \in M_{2}(\mathbb{R}). Which of the following statements are true?

  1. A.If AAᵀ = I, then DFA:M2(R)→S_A : M_{2}(\mathbb{R}) \to S is surjective.✓ Correct
  2. B.If AAᵀ = I, then DFA:M2(R)→S_A : M_{2}(\mathbb{R}) \to S need not be surjective.
  3. C.If A is invertible, then DFA:M2(R)→S_A : M_{2}(\mathbb{R}) \to S is surjective.✓ Correct
  4. D.If A is not invertible, then DFA:M2(R)→S_A : M_{2}(\mathbb{R}) \to S is surjective.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests hypothesis dropped.

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50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: Equivalent matrices are similar

More on this topic

The chapter behind this: Linear transformations, matrices and similarity — free to read

From Vector Spaces and Linear Maps › Linear transformations, matrix representation, change of basis

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