Skip to content
Part BCSIR NET December 2023indefinite-implies-isotropic

Indefinite implies isotropic

Let (−, −) be a symmetric bilinear form on R2\mathbb{R}^{2} such that there exist nonzero v,w∈R2v, w \in \mathbb{R}^{2} with (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2×2 real symmetric matrix representing this form in the standard basis. Which one of the following statements is true?

  1. A.A2=0A^{2} = 0.
  2. B.rank A = 1.
  3. C.rank A = 0.
  4. D.There exists u∈R2,u≠0u \in \mathbb{R}^{2}, u \ne 0 such that (u, u) = 0.✓ 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 converse assumed.

See pricing

50 are analysed free — try those first.

The trap it tests

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this

Related counterexample: Every real symmetric matrix is positive definite if det > 0

More on this topic

The chapter behind this: Quadratic forms, signature and definiteness — free to read

From Inner Product Spaces and Forms › Quadratic forms, positive definiteness, Sylvester's law

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

ShareWhatsAppTelegram