Skip to content
Start free
Part CCSIR NET December 2024largest-block-and-block-count-determine-six-but-not-seven

Largest block and block count determine six but not seven

Let n be a positive integer and A, B be n × n complex matrices such that the minimal polynomials of A and B are the same. Which of the following conditions ensure that A is similar to B?

  1. A.n = 3 and the characteristic polynomials of A and B are the same.✓
  2. B.n = 4 and A has two distinct eigenvalues.
  3. C.n = 5 and A has some eigenvalue for which the dimensions of the eigenspaces of A and B are the same.
  4. D.n = 6 and A has only one eigenvalue and for this eigenvalue the dimensions of the eigenspaces of A and B are the same.✓

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.

See pricing

50 are analysed free — try those first.

Related counterexample: Every real matrix has a Jordan form over ℝ

More on this topic

The chapter behind this: Jordan canonical form — free to read

From Eigenvalues and Canonical Forms › Jordan canonical form

ShareWhatsAppTelegram