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Let $A$ be an $n \times n$ matrix with rank $k$. Consider the following statements:

  1. If $A$ has real entries, then $AA^{t}$ necessarily has rank $k$
  2. If $A$ has complex entries, then $AA^{t}$ necessarily has rank $k$.

Then

  1. (i) and (ii) are true
  2. (i) and (ii) are false
  3. (i) is true and (ii) is false
  4. (i) is false and (ii) is true
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