# Matrix-Trace and Conjugate Transpose (Multiple Choice)

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.

Only 2 is false. You can easily decide on all four by using $$\text{Tr}(A^*A)=\sum_{j=1}^n\sum_{k=1}^n|A_{kj}|^2.$$
• If $|A_{kj}|\geq1$ for all $k,j$, then adding $n^2$ of those numbers will be at least $n^2$. – Martin Argerami May 4 '14 at 14:46