I have two positive semidefinite matrices, $A$ and $B$, of size $((N-1) \times (N-1))$.
I know that for all $i$ and $j$ in $\{1,...N-1\}$, $|A_{i,j}-B_{i,j}|<k$ for some positive real constant $k$ and $|A_{i,j}|<l$ and $|B_{i,j}|<l$ for some positive real constant $l$. How can I place on the below expression an upper bound in terms of $k$, $l$ and $N$?
$$|\det(A)-\det(B)|$$
I.e. I'm looking for an expression in terms of $N$ and $k$, $C_{n,k,l}$, such that:
$$|\det(A)-\det(B)|\leq C_{n,k,l}$$
I believe such an expression exists as I am working through a proof which does something similar but gives no explanation as to how it achieved it. Going by what is in the proof, it seems as though the answer should come to something similar to:
$C_{n,k,l}=k2^{N}N(N!)(l+1)^{N-1}/2$
The proof I am working through says this result should be obvious but I am unsure why.
