# Let $A$ and $B$ be real matrix such that $A+iB$ is non singular show that there exist $t \in \mathbb{R}$ such that $A+tB$ is non singular

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?

• What did you try? – José Carlos Santos Aug 3 '18 at 16:55
• 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? – Arthur Aug 3 '18 at 16:57
• $p(t)=det(A+tB)$ 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 $t$. – user580373 Aug 3 '18 at 17:04
• @spiralstotheleft / That is, it cannot be $0$ for all real $t$. – DanielWainfleet Aug 4 '18 at 1:34
• I wonder if there is a geometric proof of this? – copper.hat Aug 6 '18 at 2:20