Invertibility of Complex Matrices: Investigating f(c)=det(A+cI)

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Homework Statement


Does there exist a matrix A such that the entire family A + cI is invertible for all complex numbers c?


Homework Equations


A matrix is invertible if det(A) != 0


I really have no clue where to go with this problem. Any hints or suggestions would be greatly helpful, even if you can't give an answer to the problem.
 
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What kind of function is f(c)=det(A+cI)?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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