Understand the problem

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 and 2 are true
• 1 and 2 are false
• 1 is true and 2 is false
• 1 is false and 2 is true
Source of the problem
TIFR 2019 GS Part A, Problem 7
Linear algebra
Easy

Linear Algebra by Stephen H. Friedberg

Do you really need a hint? Try it first!

$$x$$ is a null vector of $$AA’$$ implies that $$x$$ is a null vector of $$A’$$ Let $$q_i$$ be the null vector of $$AA’$$ i.e $$AA’q_i=0 \Rightarrow q_i’AA’q_i=0 \Rightarrow ||A’q_i||_2=0 \Rightarrow A’q_i=0$$ [$$A’q_i \in \Bbb R^n$$ ] $$q_i$$ is a null vector of $$A’$$
$$x$$ is a null vector of $$A’$$ implies that \$$$x$$ is a null vector of $$AA’$$
$$Rank(AA’)=Rank(A’)=Rank (A)$$
Consider $$A= \begin{pmatrix} 1 & i \\ 0 & 0 \\ \end{pmatrix}$$ then $$AA’= \begin{pmatrix} 0 & 0 \\ 0& 0 \\ \end{pmatrix}$$

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