Question about inverse of matrix

[Carl Rowan]

Homework Statement

[/B]
Given this matrix
$\begin{bmatrix}As+B \\ C \end{bmatrix}$
which is invertible and $A$ has full row rank. I would like to show that its inverse has no terms with $s$ or higher degree if
$\begin{bmatrix}A \\ C \end{bmatrix}$
is invertible.

Homework Equations

The Attempt at a Solution

The only thing I have concluded is that you can manipulate it like this:

$\begin{bmatrix}I & 0 \\ 0 & sI \end{bmatrix} \begin{bmatrix}As+B \\ C \end{bmatrix}=\begin{bmatrix}As+B \\ Cs \end{bmatrix}=s\begin{bmatrix}A \\ C \end{bmatrix}+\begin{bmatrix}B \\ 0 \end{bmatrix}$
where each $I$ is conformable with the matrix product. Not sure how to take this any further, though. I think my main problem is that I don't know how to use the rank property of $A$.

 

[mfb]

What can you say about the determinant of $\begin{bmatrix}As \\ C \end{bmatrix}$ in terms of powers of s?
How can you use the determinant (of the original matrix) to find entries of the full inverse matrix?

 

[Carl Rowan]

Thank you for your hints, pushed me into the right direction and helped me prove the statement!