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Math Help - How To Get The Inverse In A 3x 3 Matrix?

  1. #1
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    How To Get The Inverse In A 3x 3 Matrix?

    A = [3x3] Matrix =
    -1 3 -1
    2 -2 3
    -1 1 2


    How Do We Get (A)^(-1) (The inverse of A)?

    I would show an attempt at the solution, however, I am clueless about this.
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  2. #2
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    Try this: append the matrix I to A thus:

    [A|I]=\left[<br />
\begin{array}{rrr|rrr}<br />
-1 &3 &-1 &1 &0 &0\\<br />
2 &-2 &3 &0 &1 &0\\<br />
-1 &1 &2 &0 &0 &1<br />
\end{array}\right].

    Now perform on this entire matrix the row operations necessary to get to reduced row echelon form; that is, you take A\to I. Then you'll have

    [A|I]\to[I|A^{-1}].

    All of this assumes, of course, that A is invertible.

    Make sense?
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  3. #3
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    If Ackbeet's method is a bit tricky then try using some software like MATLAB to solve it for you.
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  4. #4
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    Quote Originally Posted by AlphaRock View Post
    A = [3x3] Matrix =
    -1 3 -1
    2 -2 3
    -1 1 2


    How Do We Get (A)^(-1) (The inverse of A)?

    I would show an attempt at the solution, however, I am clueless about this.
    \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A

    Which is why singular matrices don't have an inverse.
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  5. #5
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    Quote Originally Posted by dwsmith View Post
    \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A
    Hey dw, is this correct? It may be too early in the mornig for me but it has sent my head spinning.
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  6. #6
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    Quote Originally Posted by dwsmith View Post
    \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A

    Which is why singular matrices don't have an inverse.
    Technically, you need to have the transpose of the cofactor matrix there, not A. As pickslides pointed out.
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  7. #7
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    Quote Originally Posted by Ackbeet View Post
    Technically, you need to have the transpose of the cofactor matrix there, not A. As pickslides pointed out.
    I realized that when I read his post. 50% was right though!
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  8. #8
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    Quote Originally Posted by dwsmith View Post
    I realized that when I read his post. 50% was right though!
    Right-ho.
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  9. #9
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    Quote Originally Posted by AlphaRock View Post
    A = [3x3] Matrix =
    -1 3 -1
    2 -2 3
    -1 1 2


    How Do We Get (A)^(-1) (The inverse of A)?

    I would show an attempt at the solution, however, I am clueless about this.
    Don't you have class notes or a textbook that explain what to do? (And if you don't, why are you attempting this question?)
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