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Part CCSIR NET December 2024the-two-matrices-represent-one-map-so-t-is-similar-to-t-squared

The two matrices represent one map so t is similar to t squared

Let T:C2→C2T : \mathbb{C}^{2} \to \mathbb{C}^{2} be aC−a \mathbb{C}-linear transformation. For any ordered basis ℬ of C2\mathbb{C}^{2}, let [T]ℬ denote the matrix of T with respect to ℬ. Suppose that ℬ1_{1} and ℬ2_{2} are two ordered bases of C2\mathbb{C}^{2} such that the matrix [T](ℬ1)_{1}) is upper-triangular and [T]_(ℬ2)=([T](_{2}) = ([T]_(ℬ1))2_{1}))^{2}. Which of the following statements are FALSE?

  1. A.The characteristic polynomial of T can be x2+x+1x^{2} + x + 1.
  2. B.The characteristic polynomial of T can be x(x − 1).
  3. C.The minimal polynomial of T can be x2x^{2}.✓ Correct
  4. D.The characteristic polynomial of T can be x(x−e(2πi/3))x(x - e^(2\pi{}i/3)).✓ 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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