If I try to get an inverse of a matrix:

a | b | c b | d | 0 c | 0 | 0 Inverse[{{a, b, c}, {b, d, 0}, {c, 0, 0}}] 

I get the answer:

{{0, 0, c^(-1)}, {0, d^(-1), -(b/(c d))}, {c^(-1), -(b/(c d)), (b^2 - a d)/(c^2 d)}} 

which is

 0 | 0 | 1/c 0 | 1/d | -b/(c d) 1/c | -b/(c d) | (b^2-a d)/(c^2 d)) 

Now, suppose I want to the inverse of a matrix

 A | b | c b' | d | 0 c' | 0 | 0 

where say $A$ is a symbol matrix of size N x N, $b$ is a vector of size N x 1, $c$ is a vector of size N x 1, $d$ is a scalar of size 1 x 1.

How can I write a formula for Inverse of the above matrix such that I get answers back in terms that use matrix inverse instead of scalar division, e.g. say $(cc')^{-1}c$ instead of $1/c$, and other forms like $A^{-1}$ instead of $1/A$ etc.