Prove that all the eigenvalues of $A$ are real.(Gershgorin 's Theorem) The question and its answer are given below:




And this is the Greshgorin theorem:
 
My questions are:
1- I do not understand why a consequence of Greshgorin's theorem is that if a circle is disjoint then it contains 1 eigenvalue. could anyone explain this for me, please?
2- Why in a characteristic polynomial with real coefficients the complex roots occur in pairs?
3- Why in the last paragraph the $\lambda $ and $\bar{\lambda}$ inside the disk should be the same? and why these leads to all the eigenvalues are real? could anyone help me in answering those questions, please?

The disjointness of the circles is an additional constraint imposed by your question. It is not a general feature of real matrices.
