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Let $A$ and $B$ be real matrices, with $A+iB$ non-singular. I need to show that there exist a real number $t$ such that $ A+tB $ is non-singular.

I don't have any idea how I can approach this question... could I please get a hint?

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  • $\begingroup$ What did you try? $\endgroup$ – José Carlos Santos Aug 3 '18 at 16:55
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    $\begingroup$ I don't believe that you don't have any idea. You know more than you think. The question asks about non-singular matrices. What do you know about those? How would you most easily tell whether a matrix is singular or not? Have you tried to apply that here? $\endgroup$ – Arthur Aug 3 '18 at 16:57
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    $\begingroup$ $p(t)=det(A+tB)$ is a polynomial with real coefficients. It is a non-zero polynomial, because $p(i)\neq0$. Therefore, it cannot be zero for all values of $t$. $\endgroup$ – user580373 Aug 3 '18 at 17:04
  • $\begingroup$ @spiralstotheleft / That is, it cannot be $0$ for all real $t$. $\endgroup$ – DanielWainfleet Aug 4 '18 at 1:34
  • $\begingroup$ I wonder if there is a geometric proof of this? $\endgroup$ – copper.hat Aug 6 '18 at 2:20

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