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Part CCSIR NET December 2023invertibility-from-matrix-identity

Invertibility from matrix identity

Let X, Y be two n × n real matrices such that XY =X2+X+I= X^{2} + X + I. Which of the following statements are necessarily true?

  1. A.X is invertible.✓ Correct
  2. B.X + I is invertible.
  3. C.XY = YX.✓ Correct
  4. D.Y is invertible.

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: Same characteristic polynomial ⇒ similar

More on this topic

The chapter behind this: Characteristic vs minimal polynomial — what each tells you — free to read

From Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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

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