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Part BCSIR NET December 2025solve-trace-and-determinant-for-the-other-two-eigenvalues-before-picking-an-option

Solve trace and determinant for the other two eigenvalues before picking an option

Let A be a 3×3 matrix over complex numbers with trace 1 and determinant 1. Suppose, further, that one of the eigenvalues of A is 1. Which of the following statements is necessarily true?

  1. A.The characteristic polynomial of A has repeated roots.
  2. B.Every eigenvalue of A has absolute value 1.✓
  3. C.A does not have any eigenvalue on the imaginary axis.
  4. D.A2A^{2} is the identity matrix.

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.

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50 are analysed free — try those first.

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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