Let $A$ be an $n\times n$ matrix of complex numbers. Let $B$ be the matrix $A$ with the last row and column removed. Is there a relation between the characteristic polynomials of $A$ and $B$? More generally, how does the characteristic polynomial of a matrix change with respect to the action of taking a principal minor?
We can ask the same questions for eigenvalues/eigenvectors, which are closely related with the char polynomial. Intuitively, I feel that the eigenvalues of $B$ are also the eigenvalues of $A$.
Your intuition is completely wrong, as simple examples will show.
The best way I can relate the characteristic polynomials is this. Write $$ A = \pmatrix{B & c\cr d & e\cr} $$ where $B$ is $(n-1)\times (n-1)$, $c$ is $(n-1) \times 1$, $d = 1 \times (n-1)$, and $e$ is a scalar. Then
$$ \det(A-\lambda I) = \det(B-\lambda I) (e - \lambda) - d \; \text{adj}(B-\lambda I) c$$ where $\text{adj}$ is the classical adjoint. In general the term $d \; \text{adj}(B-\lambda I) c$ will be a polynomial of degree $n-2$.