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Suppose $m$ and $n$ are positive integers, $m \neq n$, and $A$ is an $m \times n$ matrix with real entires. Consider the following statements.

  1. $\operatorname{rank}\left(A A^{T}\right)=\operatorname{rank}\left(A^{T} A\right)$
  2. $\operatorname{det}\left(A^{T} A\right)=\operatorname{det}\left(A A^{T}\right)$
  3. $\operatorname{Trace}\left(A A^{T}\right)=\operatorname{Trace}\left(A^{T} A\right)$

Which of the above statements is true for all such $A?$ 

  1. Only (i)
  2. Only (ii)
  3. Only (iii)
  4. (i) and (iii)
  5. None of them
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