Find (nontrivial) matrix $A\in M_n$ such, that only small error in its element has small influence on inverse matrix, it means that this matrix will be different from $A^{-1}$ only little bit. Then find (nontrivial) matrix $B\in M_n$ such, that only small error in its element has bigg influence on inverse matrix, it means that this matrix will be different from $B^{-1}$ a lot.
I have already find these matrices. Consider matrix real matrix $A$ $$A=\begin{pmatrix}0 & 1\\ 1 & 1\end{pmatrix},$$ then the inverse matrix $A^{-1}$ is $$A^{-1}=\begin{pmatrix}-1 & 1\\ 1 & 0\end{pmatrix}.$$
Now consider matrix $A+\Delta A$ in the form $$A+\Delta A=\begin{pmatrix}\frac{1}{1000} & 1\\ 1 & 1\end{pmatrix},$$ then the inverse matrix $A^{-1}$ is $$(A+\Delta A)^{-1}=\begin{pmatrix}-\frac{1000}{999} & \frac{1000}{999}\\ \frac{1000}{999} & -\frac{1}{999}\end{pmatrix}.$$
It means that $A^{-1}$ and $(A+\Delta A)^{-1}$ differ only little bit.
But when I consider matrix $B$
$$B=\begin{pmatrix}\frac{1001}{1000} & 1\\ 1 & 1 \end{pmatrix},$$ then the inverse matrix $B^{-1}$ is $$B^{-1}=\begin{pmatrix}1000 & -1000\\ -1000 & 1001\end{pmatrix}.$$
Now consider matrix $B+\Delta B$ in the form $$B+\Delta B=\begin{pmatrix}\frac{1002}{1000} & 1\\ 1 & 1 \end{pmatrix},$$ then the inverse matrix $A^{-1}$ is $$(B+\Delta B)^{-1}=\begin{pmatrix}500 & -500\\ -500 & 501\end{pmatrix}.$$
It means that $B^{-1}$ and $(B+\Delta B)^{-1}$ differ a lot.
But furthermore what I should do, it is not only to find matrices, but also formally justify it for any matrix. The hint is that I should use knowledges about decompositions of matrices and about condition number, also I can use this relation
$$\frac{||(A+\Delta A)^{-1}-A^{-1}||}{||A^{-1}||}\leq \kappa(A)\frac{||\Delta A||}{||A||},$$
where $\kappa(A)$ is condition number of matrix $A$. I think that it can be something with regularity and singularity of matrix. My matrix $A$ is "strong" regular and difference between these two inverse matrices was small. On the other hand the matrix $B$ was near to singular and the difference between inverse matrices was bigg.
I have not idea how to deal with formally and more generally. Any help will be appreciated. Thank you very much.
I think your inequality is only correct to fist order in $\Delta A$. You really need the condition number of the problem of inverting $A$. Using calculus (see here: https://en.wikipedia.org/wiki/Condition_number#Several_variables) and Hahn Banach theorem, the condition number can be shown to be $\kappa(A)$. This means your inequality is actually sharp, at least to first order. That means there are small perturbations $\Delta A$ for which the inequality is almost an equality. Now you need to determine when $\kappa(A)$ is "large".
Let $\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_n > 0$ be the singular values of $A$. Using the $2$-norm, we have $\lVert A \rVert = \sigma_1$, $\lVert A^{-1} \rVert = \frac{1}{\sigma_n}$. Thus $\kappa(A) = \frac{\sigma_1}{\sigma_n}$. Thus if $\sigma_n$ is small and $\sigma_1$ is large, then $k(A)$ will be large. So try $A = \begin{bmatrix} 1000 & 0 \\ 0 & 0.1 \end{bmatrix}$.