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 $B = A - I$?

There is no simple answer.  $P(\lambda) = \det(A-\lambda I)$ is the characteristic polynomial of $A$.  This is a polynomial of degree $n$: the coefficient of $\lambda^n$ is $(-1)^n$ and the coefficient of $\lambda^0$ is $\det(A)$.   The coefficients of other powers of $\lambda$ are various functions of the entries of $A$. $\det(A+I)$ and $\det(A-I)$ are $P(-1)$ and $P(1)$; that's about all there is to say about them.
