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Part CCSIR NET December 2024each-determinant-is-raised-to-the-dimension

Each determinant is raised to the dimension

Let M2(R)M_{2}(\mathbb{R}) denote the R−\mathbb{R}-vector space of 2 × 2 matrices with real entries. Let A = [[1, 2], [0, 3]] and B = [[−1, 0], [1, 5]]. Define a linear transformation T:M2(R)→M2(R)T : M_{2}(\mathbb{R}) \to M_{2}(\mathbb{R}) by T(X) = AXBᵗ, where Bᵗ denotes the transpose of the matrix B. Which of the following statements are true?

  1. A.det(T) = 225✓ Correct
  2. B.det(T) = −225
  3. C.Trace(T) = 16✓ Correct
  4. D.Trace(T) = −16

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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: AB = I ⇒ BA = I for all matrices

More on this topic

The chapter behind this: Determinants, trace and block matrices — free to read

From Determinants and Matrix Tricks › Determinants, trace, block matrices, rank inequalities

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