Relationship between $\det(\bf{A}+\bf{B})$ and  $\det(\bf{A})$ How $\det(\bf{A}+\bf{B})$ is related with $\det(\bf{A})$ where $\bf{A}$ is either semi-definite or positive definite matrix but $\bf{B}$ is a zero diagonal indefinite matrix. $\det(\cdot)$ denotes the determinant and all matrices are square. Off-diagonal elements of $\bf{B}$ are all positive. All elements of $\bf{A}$ are positive.

$A=\pmatrix{2&-1\cr-1&2\cr}$ is positive definite with determinant 3. 
$B=\pmatrix{0&b\cr b&0\cr}$ has zero diagonal and is indefinite. The determinant of $A+B$ is $4-(b-1)^2$ which can be greater than, less than, or equal to 3. This suggests that in general you can't say very much. 
