TITLE: how does addition of identity matrix to a square matrix changes determinant?
QUESTION [9 upvotes]: 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 $B = A - I$?

REPLY [3 votes]: I would like to contribute the following special case which I encountered when I was looking for an answer and stumbled upon this thread. 
If $A=uv^\intercal$, then 
$\det(I+A) = 1+u^Tv $