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Part CCSIR NET June 2024algebraic-closure-supplies-the-eigenvalue

Algebraic closure supplies the eigenvalue

Let M5(C)M_{5}(\mathbb{C}) be the complex vector space of 5 × 5 matrices with entries in C\mathbb{C}. Let V be a non-zero subspace of M5(C)M_{5}(\mathbb{C}) such that every non-zero A ∈ V is invertible. Which among the following are possible values for the dimension of V?

  1. A.1✓ Correct
  2. B.2
  3. C.3
  4. D.5

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 base field or ring.

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50 are analysed free — try those first.

The trap it tests

Base field or ring

The answer changes with the field or ring you are working over.

Drill statements like this

Related counterexample: Same characteristic polynomial ⇒ similar

More on this topic

The chapter behind this: Characteristic vs minimal polynomial — what each tells you — free to read

From Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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