If $\mbox{tr}(AA^*) < n^2$ then $|a_{ij}| < 1$ for some $i,j$

Question

Let $A = (a_{ij})$ be an $n \times n$ complex matrix and let $A^*$ denote the conjugate transpose of $A$. Which of the following statements are necessarily true?

1. If $\mbox{tr}(AA^*) \neq 0$ then $A$ is invertible.
2. If $\mbox{tr}(AA^*) < n^2$ then $|a_{ij}| < 1$ for some $i,j$
3. If $\mbox{tr}(AA^*) > n^2$ then $|a_{ij}| > 1$ for some $i,j$

---

My attempt

We know that

$$ \text{Tr}(AA^*)=\sum_{i=1}^n\sum_{j=1}^n|a_{ij}|^2. $$

If $A$ is nilpotent matrix then $tr(AA^*) \neq 0$ but $detA = 0$. So option 1 is not true.

If $|a_{ij}| < 1$ for some $i,j$. Taking summation on both sides we get

$$\sum_{i=1}^n\sum_{j=1}^n|a_{ij}|^2 < n^2$$

I'm not sure that I'm on right path. Please help me.