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Part BCSIR NET June 2023char-vs-min-polynomial

Char vs min polynomial

Let T be a linear operator on R3\mathbb{R}^{3}. Let f(X)∈R[X]f(X) \in \mathbb{R}[X] denote its characteristic polynomial. Consider the following statements. (a) Suppose T is non-zero and 0 is an eigenvalue of T. If we write f(X) = X·g(X) in R[X]\mathbb{R}[X], then the linear operator g(T) is zero. (b) Suppose 0 is an eigenvalue of T with at least two linearly independent eigenvectors. If we write f(X) = X·g(X) in R[X]\mathbb{R}[X], then the linear operator g(T) is zero. Which of the following is true?

  1. A.Both (a) and (b) are true.
  2. B.Both (a) and (b) are false.
  3. C.(a) is true and (b) is false.
  4. D.(a) is false and (b) is true.✓ Correct

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 invariants don't determine.

See pricing

50 are analysed free — try those first.

The trap it tests

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

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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