Invariants don't determine

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

Would you have fallen for these?

Each is a real option from a previous-year paper. Read it cold and decide before opening it.

• Option A

  The eigenspace of A for the eigenvalue 0 is two-dimensional.

  Why this is wrong, and why it looks right

  False — this confuses algebraic with geometric multiplicity. $T^2$ dividing p means 0 has algebraic multiplicity at least 2, but the eigenspace can still be one-dimensional if the Jordan block has size 2. SymPy takes A = [[0,1,0],[0,0,0],[0,0,1]], whose characteristic polynomial is ${\lambda }^3-{\lambda }^2={\lambda }^2(\lambda -1)$, and finds rank 2 — so the 0-eigenspace has dimension 3 − 2 = 1, not 2.

  From this question

  Let A be a 3×3 real matrix whose characteristic polynomial p(T) is divisible by $T^2$. Which of the following statements is true?

  See the full question

  Part B · june 2023 · free

• Option D

  A is diagonalizable.

  Why this is wrong, and why it looks right

  False, refuted by the same matrix: it has a size-2 Jordan block at 0, so it is not diagonalizable. Diagonalizability would need the geometric multiplicity of 0 to reach 2, which is exactly what (A) wrongly assumes.

  From this question

  Let A be a 3×3 real matrix whose characteristic polynomial p(T) is divisible by $T^2$. Which of the following statements is true?

  See the full question

  Part B · june 2023

• Option D

  S is in bijection with the power set of S.

  The analysis of this one comes with the PYQ pack.

  Part B · june 2023

• Option B

  The minimal polynomial of A is $x^2(x-1)(x-2\pi i{)}^3$.

  The analysis of this one comes with the PYQ pack.

  Part C · december 2023

• Option C

  The dimensions of the eigenspaces for $0,1,2\pi i$ are 2, 1, 3 respectively.

  The analysis of this one comes with the PYQ pack.

  Part C · december 2023

• Option B

  Suppose that for every vector $v\in {\mathbb{R}}^5$ there exists n such that $T^{n}v=0$. Then $T^2$ must be zero.

  The analysis of this one comes with the PYQ pack.

  Part C · december 2023

• Option D

  countably infinite

  The analysis of this one comes with the PYQ pack.

  Part B · december 2023

• Option A

  Quadratic forms (a) and (b) are equivalent.

  The analysis of this one comes with the PYQ pack.

  Part C · june 2023

Other mistakes that cost marks