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Thread: Cramer Rule and Vector Spaces

  1. #1
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    Cramer Rule and Vector Spaces

    1) How do you find A (by hand) if an adjoint of A 4 by 4 matrix is given you?

    For example:
    Adj A equals

    2 0 0 0
    0 2 1 0
    0 4 3 2
    0 -2 -1 2

    what is A?

    I know that A (adj. A) = (det. A) I
    and I tried to find A by finding the Adj. A of the given Adj. of A and I got the weird numbers.

    The answer is suposed to give the following matrix:
    A equals
    1 0 0 0
    0 4 -1 1
    0 -6 2 -2
    0 1 0 1

    How do you get to that? I had asked this several days ago and the links I received sort of helped but I got confused again. I tried to solve this twice and its frustrating that I am not getting the answer. Please if anyone could explain this with baby steps I don't mind at all I just want is to understand it. Thanks

    2) Let R denote the set of real numbers. Define scalar multiplication by
    a x= a *x (the usual multiplication of real numbers)
    and define the addition, denoted , by
    x y = max (x, y) (the maximum of the two numbers)
    Is R a vector space with these operations? Prove your answer.
    ( I have no idea how to solve this one either)
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  2. #2
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    1) If A is an n by n matrix then det(adj A) = (det A)^(n-1). Also, if det A is not zero then A = (det A)(adj A)^(-1). For this example,

    det(adj A) =\begin{vmatrix}2&0&0&0\\0&2&1&0\\0&4&3&2\\0&-2&-1&2\end{vmatrix} = 8 = 2^3, and so det A = 2. You then have to find the inverse of adj A and multiply it by 2, in order to get A.

    2) A vector space has to have a zero element. If R with the given operation is a vector space then there must exist an element O of R such that x O = x for every x in R. Think about what that says about O. (Of course, O need not be the usual 0. In fact, it would have to have very unusual properties.)
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by googoogaga View Post
    2) Let R denote the set of real numbers. Define scalar multiplication by
    a x= a *x (the usual multiplication of real numbers)
    and define the addition, denoted , by
    x y = max (x, y) (the maximum of the two numbers)
    Is R a vector space with these operations? Prove your answer.
    ( I have no idea how to solve this one either)
    You go through the list of axioms defining a vector space showing that
    does or does not satisfy these axioms over the reals.

    The one you will have a problem with the one Opalg pointed out, in fact you
    will still have problems if you work with extended reals as you will need

    +infty+(-infty)=0,

    but this combination of the two infinities is usually left undefined.

    RonL
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  4. #4
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    sorry opalg I thought I undestood the first problemm. I got stuck again. How do you find the inverse of the given adjoint of A I am lost completely eveything you wrote made sense until I started working on it again. How do you get to the matrix I have to see it in order to understand it. I thought I did with the explanations you gave me but I am getting nowhere. Please help...
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  5. #5
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    Quote Originally Posted by googoogaga View Post
    How do you find the inverse of the given adjoint of A
    Same way as you would find the inverse of any other matrix. And that depends on which method you normally use for finding inverses. Since this is a problem about adjoint matrices, maybe you know the formula A^{-1} = \frac{\mathrm{adj}(A)}{\mathrm{det}(A)}? In that case, you can find the inverse of adj(A) by calculating adj(adj(A)). In fact, (\mathrm{adj}(A))^{-1} = \frac{\mathrm{adj}(\mathrm{adj}(A))}{\mathrm{det}(  \mathrm{adj}(A))}.
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