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2x2 matrix inverse formula

  1. Sep 24, 2008 #1
    I need to find the inverse of a 2x2 matrix [a b ; c d] using Gauss-Jordan elimination.

    I am halfway there but I'm stuck on the algebra because it gets really messy. Could anyone possibly do it step by step?
     
  2. jcsd
  3. Sep 24, 2008 #2

    gabbagabbahey

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    Why don't you show us what you've got so far?
     
  4. Sep 24, 2008 #3
    sure:

    [ a b ; c d | 1 0 ; 0 1 ] -->
    [ a b ; (ac/c) (ad/c) | 1 0 ; 0 (a/c) ] -->
    [ a b ; 0 ((ad/c)/c) -b | -1 (a/c) ] -->
    ...

    here's where i'm a little stuck. i'm bad at keeping track of every variable...i think i miss something along the way because of the messy algebra.
     
  5. Sep 24, 2008 #4

    gabbagabbahey

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    Assuming that your last line is supposed to be:
    [tex]\begin{pmatrix} a & b &1 & 0 \\ 0 & \frac{ad}{c}-b & -1 & \frac{a}{c} \end{pmatrix}[/tex]

    then your doing fine so far. what is your next step?
     
  6. Sep 24, 2008 #5
    here it is:

    [ (a(((ad/c)-b)/b) (((ad/c)-b)/b) ; 0 ((ad/c) -b) | (((ad/c)-b)/b) 0 ; -1 (a/c) ]

    look good?
     
  7. Sep 24, 2008 #6

    gabbagabbahey

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    Are you multiplying the top row by (ad/c-b)/b ? If so, you should get:
    [tex]
    \begin{pmatrix} \frac{a(\frac{ad}{c}-b)}{b} & (\frac{ad}{c}-b) &\frac{(\frac{ad}{c}-b)}{b} & 0 \\ 0 & \frac{ad}{c}-b & -1 & \frac{a}{c} \end{pmatrix}
    [/tex]
     
  8. Sep 24, 2008 #7
    ah right, so the next step is:

    [ (a(((ad/c)-b)/b) - (((ad/c)-b)) 0 ; 0 ((ad/c) -b) | ((((ad/c)-b)/b) - (ad/c) -b) 0 ; -1 (a/c) ]

    it's messy this way...sorry.
     
  9. Sep 24, 2008 #8

    gabbagabbahey

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    Wouldn't the step be to subtract the bottom row from the top row to get:
    [tex]\begin{pmatrix} \frac{a(\frac{ad}{c}-b)}{b} & 0 &\frac{(\frac{ad}{c}-b)}{b}+1 & \frac{-1}{c} \\ 0 & \frac{ad}{c}-b & -1 & \frac{a}{c} \end{pmatrix}=\begin{pmatrix} \frac{a(\frac{ad}{c}-b)}{b} & 0 &\frac{ad}{bc} & \frac{-1}{c} \\ 0 & \frac{ad}{c}-b & -1 & \frac{a}{c} \end{pmatrix}[/tex]
     
  10. Sep 24, 2008 #9
    alright, so now we have a matrix with zeros along the anti-diagonal. the inverse doesn't equal the inverse given by the 2x2 inverse formula. what went wrong?
     
  11. Sep 24, 2008 #10

    gabbagabbahey

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    You still have to set the diagonal elements to 1: simply multiply the top row by b/(a(ad/c-b)) and the bottom row by 1/(ad/c-b)
     
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