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Question about inverse of matrix

  1. Jan 24, 2017 #1
    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. jcsd
  3. Jan 24, 2017 #2

    mfb

    User Avatar
    2016 Award

    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?
     
  4. Jan 27, 2017 #3
    Thank you for your hints, pushed me into the right direction and helped me prove the statement!
     
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