I need to calculate a matrix $A$ (at least some elements of it, see below) as defined by the following equation

$$ A=B(\mathbb{1}-B)^{-1} $$

where B is a square matrix of dimension $N$ and $\mathbb{1}$ is $N \times N$ identity matrix.

Inspired by this post:

https://www.johndcook.com/blog/2010/01/19/dont-invert-that-matrix/

I was wondering if I really need to invert $\mathbb{1}-B$ in my case, of if there's some easier way. Keep in mind that:

• $N$ in my case is quite large, its order of magnitude can be ten thousand.

• I don't need to know the full matrix $A$, I just need a few elements in the upper left corner, let's say $A_{00}$, $A_{01}$, $A_{11}$, $A_{02}$, $A_{12}$, $A_{22}$ would be perfect.

I need to calculate a matrix $A$ (at least some elements of it, see below) as defined by the following equation

$$ A=B(\mathbb{1}-B)^{-1} $$

where B is a square matrix of dimension $N$ and $\mathbb{1}$ is $N \times N$ identity matrix.

Inspired by this post:

https://www.johndcook.com/blog/2010/01/19/dont-invert-that-matrix/

I was wondering if I really need to invert $\mathbb{1}-B$ in my case, or if there's some easier way. Keep in mind that:

• $N$ in my case is quite large, its order of magnitude can be ten thousand.

• I don't need to know the full matrix $A$, I just need a few elements in the upper left corner, let's say $A_{00}$, $A_{01}$, $A_{11}$, $A_{02}$, $A_{12}$, $A_{22}$ would be perfect.