I would like to get a better understanding of limits and matrix inverses, specifically the relationship between:
$\lim_{k\rightarrow \infty}(\mathbf{A}^{-1})$ and $(\lim_{k\rightarrow \infty}\mathbf{A})^{-1}$ (*)
where $\mathbf{A} = \left[\begin{array}{c c} {\mathbf{B}} & \mathbf{b}\\ \mathbf{b}^{'} & k \end{array}\right]$ and $k$ is a scalar. Lets assume that $\mathbf{A}$ is invertible for all $k$.
Clearly the terms in (*) are not equivalent. However, lets say that I want to take the ratio of two elements of $\mathbf{A}^{-1}$. Say I want to evaluate:
$r = \frac{\mathbf{A}^{-1}_{i,n}}{\mathbf{A}^{-1}_{j,n}}$.
I can apply the matrix inversion lemma and get an expression of the following form:
$r = \frac{\frac{1}{k+c}{h}}{\frac{1}{k+c}{g}}$ (this form will only works for certain elements)
where $c,h,g$ are some scalars that can be calculated using the matrix inversion lemma and more importantly, $h,g$ do not depend on $k$. We see that the terms that do depend on $k$ cancel and we are left with $r = \frac{h}{g}$. From this I conclude that the ratio of certain elements of $\mathbf{A}^{-1}$ do not depend on $k$.
I'm a little unsure how to justify this rigorously. My final result for $r$ is correct but it seems that I'm taking the ratio of two elements of the inverse without regard for what $k$ is and then canceling terms. If on the other hand I assume that $k \rightarrow \infty$ before taking the inverse, I would get something very different.