I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate. 

Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. However, which values are changed is not known. In fact the distribution of changed elements are uniform over elements of the matrix.We do have bound over number of elements changes $\beta\ll n^{2}$.

Let's call the perturbation in the matrix $\Delta A$. So the matrix vector product calculated is $(A+\Delta A)p$. What I am intersted  is finding bounds for following quantity
   
   $\eta =\frac{||\Delta Ap||}{||Ap||}
 $ 

I would like this bound based on all known quantities like norms of $A$ and $p$ and conditions numbers, $\beta$ etc. It can be assumed that most of the parameters about $A$ and $p$ are known.
 
Edit -
So far I've done is  

$ \frac{||\Delta Ap||_{2}}{||Ap||_{2}}\leq\frac{||\Delta A||_{2}\,||p||_{2}}{\lambda_{n}||p||_{2}}\leq\frac{||\Delta A||_{F}}{\lambda_{n}}
$

  , Here I have used 2-norm is always less then frobinius norm and $||Ap||\geq\lambda_{n}||p||$
 . Here A is SPD matrix and $\lambda_{n}$
  is smallest eigenvalue. Now 

$ \Delta A||_{F}^{2}=\sum_{(i,j)\in E}a_{i,j}^{2} 
$ 
 

Hence 

$||\Delta A||_{F}\leq(\max_{i,j}(a_{i,j})\sqrt{\beta})\leq\lambda_{1}\sqrt{\beta}$

 . Here again E
  is the set of entries in the matrix where element has been changed. and \lambda_{1}
 is largest eigenvalue of the matrix. and $\max_{i,j}(a_{i,j})\leq\lambda_{1}$
 . Hence over all I get is $\frac{||\Delta Ap||_{2}}{||Ap||_{2}}\leq\kappa(A)\sqrt{\beta}$
 . However this is very loose bound since both the terms on right hand side are greater than one.