[Math] Matrix-Trace and Conjugate Transpose (Multiple Choice)

Question

I was trying to solve the following problem from a competitive exam paper.

Let $A=( a_{ij})$ be a nXn complex matrix and let $A^*$ denote the conjugate transpose of $A$. Then which of the following statements are necessarily true? (One or more options may be correct)

1. $A^{-1}$ exists $\Rightarrow tr(A^*A)\neq 0 $
2. $ tr(A^*A)\neq 0 \Rightarrow A^{-1} $ exists.
3. $|tr(A^*A)|<n^2\Rightarrow | a_{ij}|<1 $ for some $i,j$
4. $ tr(A^*A)= 0 \Rightarrow A = 0$

I am completely stuck.

Please help me.
Thnx.

Answer 1

Only 2 is false. You can easily decide on all four by using $$ \text{Tr}(A^*A)=\sum_{j=1}^n\sum_{k=1}^n|A_{kj}|^2. $$