Skip to content
Part BCSIR NET June 2024operator-on-matrices-not-on-vectors

Operator on matrices not on vectors

Let V be the real vector space of 2 × 2 matrices with entries in R\mathbb{R}. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where A = ((2, 0), (0, 1)). What is the characteristic polynomial of T?

  1. A.(x − 2)(x − 1)
  2. B.x2(x−2)(x−1)x^{2}(x - 2)(x - 1)
  3. C.(x−2)2(x−1)2(x - 2)^{2}(x - 1)^{2}✓ Correct
  4. D.(x2−2)(x2−1)(x^{2} - 2)(x^{2} - 1)

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 finite-dimensional intuition.

See pricing

50 are analysed free — try those first.

The trap it tests

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this

Related counterexample: Equivalent matrices are similar

More on this topic

The chapter behind this: Linear transformations, matrices and similarity — free to read

From Vector Spaces and Linear Maps › Linear transformations, matrix representation, change of basis

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

ShareWhatsAppTelegram