Let $A$ be an $n \times n$ matrix with rank $k$. Consider the following statements:
- If $A$ has real entries, then $AA^{t}$ necessarily has rank $k$
- If $A$ has complex entries, then $AA^{t}$ necessarily has rank $k$.
Then
- (i) and (ii) are true
- (i) and (ii) are false
- (i) is true and (ii) is false
- (i) is false and (ii) is true