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Math Help - Finding the inverse of a matrix with missing values

  1. #1
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    Finding the inverse of a matrix with missing values

    The question:
    Long ago, a mathematician wrote C and C^{-1} on a piece of paper. Unfortunately insects have damaged the paper and all that is left is:

    $C =<br />
\begin{array}{ccc}<br />
-2 & -1 & 1 \\<br />
1 & 2 & -1 \\<br />
* & * & * \\<br />
\end{array}$

    $C^{-1} =<br />
\begin{array}{ccc}<br />
* & 0 & -1 \\<br />
2 & * & -1 \\<br />
5 & 1 & * \\<br />
\end{array}$

    a) Find C^{-1}

    My attempt:
    I tried finding the inverse of C the usual way, by substituting the asterixes with variables, and then setting up the following:

    $C =<br />
\begin{array}{ccccccc}<br />
-2 & -1 & 1 & | & 1 & 0 & 0\\<br />
1 & 2 & -1 & | & 0 & 1 & 0\\<br />
a & b & c & | & 0 & 0 & 1\\<br />
\end{array}$

    I tried reducing this to row echelon form, but it quickly became a mess, and I was unable to completely reduce it. Is my approach correct so far? Thanks.
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  2. #2
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    Remember that the inverse of the matrix is given by 1/|C| * adj(C)

    So if we find all the cofactors of the matrix C, and then equate them to the known values of C^{-1}, we can then solve using simultaneous equations.

    Does this make sense?
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  3. #3
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    Give a symbol to each unknown value

    C = \left[\begin{matrix}-2&-1&\phantom{-}1\\ \phantom{-}1&\phantom{-}2&-1\\ \phantom{-}a&\phantom{-}b&\phantom{-}c\end{matrix}\right]

    C^{-1} = \left[\begin{matrix}\phantom{-}d&\phantom{-}0&-1\\\phantom{-}2&\phantom{-}e&-1\\ \phantom{-}5&\phantom{-}1&\phantom{-}f\end{matrix}\right].


    You should know that when you multiply a matrix by its inverse, you get the identity matrix.

    So \left[\begin{matrix}-2&-1&\phantom{-}1\\ \phantom{-}1&\phantom{-}2&-1\\ \phantom{-}a&\phantom{-}b&\phantom{-}c\end{matrix}\right] \left[\begin{matrix}\phantom{-}d&\phantom{-}0&-1\\\phantom{-}2&\phantom{-}e&-1\\ \phantom{-}5&\phantom{-}1&\phantom{-}f\end{matrix}\right] = \left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\righ  t].


    Perform the multiplication and you should be able to find the unknowns.
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  4. #4
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    Thank you soooo much! I've been stuck on this question for two days!
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