- #1

Carl Rowan

- 2

- 1

## Homework Statement

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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##.