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Math Help - det(A) where A is 3x3 matrix

  1. #1
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    det(A) where A is 3x3 matrix [SOLVED]

    If A is 3x3 with all entries 1's and 0's, what is the max value of det(A)?

    I answered it to be 1, which is wrong. Can someone help me?

    Thanks!
    Last edited by page929; November 27th 2011 at 06:45 AM.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: det(A) where A is 3x3 matrix

    Quote Originally Posted by page929 View Post
    If A is 3x3 with all entries 1's and 0's, what is the max value of det(A)?

    I answered it to be 1, which is wrong. Can someone help me?

    Thanks!
    Hint:

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  3. #3
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    Re: det(A) where A is 3x3 matrix

    A=\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i\end{bmatrix}

    Use Laplace's formula on the first row and you'll get

    \det(A)=a(ei-fh)-b(di-fg)+c(dh-eg)

    since each entry can only be either 1 or 0, just try to find the coefficients such that you are always summing and not subtracting. You will find that the answer to your question is 2.
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  4. #4
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    Re: det(A) where A is 3x3 matrix

    [1 1 0]
    [0 1 1]
    [1 0 1] maximizes the positive contributions, and minimizes the negative contributions. (seeing is believing).
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