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Part BCSIR NET December 2023sylvester-rank-inequality

Sylvester rank inequality

Which one of the following statements is FALSE?

  1. A.The product of two 2×2 real matrices of rank 2 is of rank 2.
  2. B.The product of two 3×3 real matrices of rank 2 is of rank at most 2.
  3. C.The product of two 3×3 real matrices of rank 2 is of rank at least 2.✓ Correct
  4. D.The product of two 2×2 real matrices of rank 1 can be the zero matrix.

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: 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

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