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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.
  4. D.The product of two 2×2 real matrices of rank 1 can be the zero matrix.

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The trap it tests

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this

Related counterexample: AB and BA have the same minimal polynomial

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