how does addition of identity matrix to a square matrix changes determinant? Suppose there is $n \times n$ matrix $A$. If we form matrix $B = A+I$ where $I$ is $n \times n$ identity matrix, how does $|B|$ - determinant of $B$ - change compared to $|A|$? And what about the case where $B = A - I$?

The characteristic polynomial $P_A(\lambda)$ of a matrix $A$ is defined as
$$
P_A(\lambda)=\det(A-I\lambda)
$$
Therefore, $\det(A)=P_A(0)$, while $\det(A+I)=P_A(-1)$ and $\det(A-I)=P_A(1)$.
