1. Jan 24, 2017

Carl Rowan

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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 $A$.

2. Jan 24, 2017

Staff: Mentor

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?

3. Jan 27, 2017

Carl Rowan

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