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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.

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

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  • $\begingroup$ Thank you very much for your help. I have a little doubt. Can you please clarify how option 3 is coming out to be true from your given formula? $\endgroup$ – usermath May 4 '14 at 14:30
  • $\begingroup$ If $|A_{kj}|\geq1$ for all $k,j$, then adding $n^2$ of those numbers will be at least $n^2$. $\endgroup$ – Martin Argerami May 4 '14 at 14:46
  • $\begingroup$ Thank you again.It is very much clear now. $\endgroup$ – usermath May 4 '14 at 14:48

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