Suppose that we are operating in a three-digit decimal floating point system. We want to use Gaussian Elimination with Partial Pivoting (GEPP) to find the inverse of a given invertible $2\times2$ matrix $A$.
The question is to construct an example of a non-singular matrix $A$, whose computed inverse matrix $X$ is non-singular, yet $X^{-1}$ is not close to $A$. More precisely, we want to show that for every given $A$, there does not exist a matrix $E$ such that $X^{-1}-A=E$ and $\frac{\lVert E \rVert_{\infty}}{\lVert A \rVert_{\infty}} \leq u$, where $u=10^{-2}$ is the unit round-off error. This literally means that computing inverse via GEPP is not backward stable
We know from the error analysis of GEPP that the computed solution of $Ax=b$ exactly satisfies the followings $$(A+E)x=b, \qquad \frac{\lVert E \rVert_{\infty}}{\lVert A \rVert_{\infty}} \leq \gamma \, n^3 g_n u, \qquad g_n=\frac{\underset{1\leq k\leq n-1}{\max}\,\,\underset{1\leq i, \, j \leq n}{\max} |a^{(k)}_{ij}|}{\underset{1\leq i, \, j \leq n}{\max} |a^{(1)}_{ij}|}, \tag{1}$$
where $n$ is the size of square matrix $A$, $k$ is the elmination step in GEPP, $i$ and $j$ are indices for rows and columns, $\gamma$ is a constant, and $g_n$ is called the growth factor. For computing the inverse, we should solve the following set of linear systems
$$Ax_m=e_m, \quad m=1,\dots,n\tag{2}$$
where $x_m$ is the $m$-th column of $X$, and $e_m$ is the $m$-th standard basis vector with $1$ in the $m$-th place and $0$ everywhere else. Combining $(1)$ and $(2)$, I can see that by this error analysis we are only guaranteed that $m$-th column of the computed inverse $X$ is the $m$-th column of a matrix $(A+E_m)^{-1}$, whose exact inverse $A+E_m$ is close to $A$ in the sense of infinity norm mentioned above. However, I don't have any idea on building an example required in the question. What would be a good way to construct such an example?