First version: Let $A$ and $B$ be (complex) Hankel matrix. Is it true that $\det (A+B)\neq 0$ if $\det A=0$ and $\det B\neq0$? No.
Reformulating: For which $B$ is it true that $\det (A+B)\neq 0$ if $\det A=0$?
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$\begingroup$ No this is not true. you can easily construct a $3 \times 3$ counterexample: A+B= {zeros on the skew-diagonal, and ones everywhere else}. A= {the upper left corner is 1 and everything else is zero} $\endgroup$– Daniel SoltészCommented Mar 3, 2014 at 7:01
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1$\begingroup$ Maybe post the 2nd version of the question as a new question and include something about motivation? $\endgroup$– Bjørn Kjos-HanssenCommented Mar 3, 2014 at 16:41
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$\begingroup$ I think what the 2nd version is asking is, the given condition seems to involve quantifying over all $A$; is it actually a simpler property? $\endgroup$– Bjørn Kjos-HanssenCommented Mar 3, 2014 at 22:50
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$\begingroup$ Yes, it has to be something universal, for all $A$... Actually, it is a problem to make singular matrix non-singular but for the class of Hankel matrices. $\endgroup$– UserrrCommented Mar 4, 2014 at 6:22
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