eigen value problem of the following matrix [duplicate]

Question

$$A=\begin{bmatrix} 1 & -2 & 3 & -2 \\ 1 & 1 & 0 & 3 \\ -1 & 1 & 1 & -1 \\ 0 & -3 & 1 & 1 \\ \end{bmatrix}$$

4.9 Pick out the smallest disc in the complex plane containing all the eigenvalues of $A$ from amongst the following:

1. $|z-1| \leq 7$;
2. $|z-1| \leq 6$;
3. $|z-1| \leq 4$.

How to solve the problem 4.9?

Answer 1

The characteristic polynomial of the matrix $$A:=\begin{bmatrix} 1 & -2 & 3 & -2 \\ 1 & 1 & 0 & 3 \\ -1 & 1 & 1 & -1 \\ 0 & -3 & 1 & 1 \\ \end{bmatrix}$$ is $$\det(A-x I)=x^4-4x^3+21x^2-48x+46.$$

Wolfram Alpha approximates its roots as $$0.4956 \pm 3.8331i$$ and $$1.5044 \pm 0.9033i.$$ These are approximately the eigenvalues of the matrix.

Now it's a matter of finding which is the smallest disc containing these four complex numbers.