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Inverse of a matrix

  1. Jun 9, 2017 #1
    1. The problem statement, all variables and given/known data

    Find the inverse of
    ##A = \begin{bmatrix} 1 & \dfrac12 & & \cdots && \dfrac1n
    \\\dfrac12 & \dfrac13 && \cdots && \dfrac1{n+1}
    \\ \vdots & \vdots && && \vdots
    \\ \dfrac1n & \dfrac1{n+1} && \cdots && \dfrac1{2n-1}\end{bmatrix}##

    2. Relevant equations


    3. The attempt at a solution


    I obvserved that ##A_{ij} = \dfrac{1}{i+j-1}##.

    Also I know ##I = AA^{-1}##

    So jth column of ##I## is ##A## times jth column of ##A^{-1}##

    So for ##j = 1##

    ##A \times \begin{bmatrix}A^{-1}_{11} \\ \vdots \\ A^{-1}_{n1}\end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ \vdots\\0 \end{bmatrix}##.

    Now I don't know what to do. Any clue.
     
  2. jcsd
  3. Jun 9, 2017 #2

    jedishrfu

    Staff: Mentor

    This might help:

     
  4. Jun 9, 2017 #3
    I know how to take inverse just I don't know how to do that in case of matrix like this.
     
  5. Jun 9, 2017 #4

    jedishrfu

    Staff: Mentor

    I don't see where you're confused. The procedure is the same. Is it that the answer is just wrong?
     
  6. Jun 9, 2017 #5
    Ok, I tried something,

    I did ##A_i \to A_i - \dfrac1i A_1##, where ##A_1, A_i## are the rows.

    I got,
    ##\begin{bmatrix} 1 & \dfrac12 & & \cdots && \dfrac1n
    \\ 0 & \dfrac13 - \dfrac12 && \cdots && \dfrac1{n+1} - \dfrac1n
    \\ \vdots & \vdots && && \vdots
    \\ 0 & \dfrac1{n+1} - \dfrac1n && \cdots && \dfrac1{2n-1} - \dfrac1n^2\end{bmatrix} =

    \begin{bmatrix} 1 & \dfrac12 & & \cdots && \dfrac1n
    \\ 0 & \dfrac1{12} && \cdots && \dfrac1{n+1} - \dfrac1n
    \\ \vdots & \vdots && && \vdots\\
    0 & \dfrac{i -1}{2i(i + 1)} && && \vdots\\
    \vdots & \vdots &&&& \vdots \\
    \\ 0 & \dfrac{n-1}{2n(n+1)} && \cdots && \dfrac1{2n-1} - \dfrac1n^2\end{bmatrix}
    ##

    See it is very messy and I don't know what to do now.
     
  7. Jun 9, 2017 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Google "inverse of special matrix".
     
  8. Jun 9, 2017 #7
    Can you provide the link to the site, I searched the first page of Google but nothing matches.
     
  9. Jun 9, 2017 #8

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Have you looked at all the other articles? I found several, just by searching as I suggested to you.
     
  10. Jun 10, 2017 #9
    Yes I have looked at each and every article on the first page.
     
  11. Jun 10, 2017 #10

    StoneTemplePython

    User Avatar
    Gold Member

    What's the purpose of this exercise? This is clearly a very special instance of a Hankel matrix which has its own name...
     
  12. Jun 10, 2017 #10

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    The Wikipedia article has all you need. Look at the entry for "Cauchy Matrix".
     
  13. Jun 10, 2017 #11

    WWGD

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    Science Advisor
    Gold Member

    Or, just work a few small cases, like ## 2 \times 2 , 3\times 3 ## and come up with an educated guess.
     
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