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Math Help - Determinant calculation. 4 Questions.

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Determinant calculation. 4 Questions.

    Need to calculate these matrix determinants without using any calculation programs, only by definition!

    Thanks a lot to whom accept this challenge!


    1. Above C


    0 , 3i , 1
    2-i , 3 , 1
    -i , 0 , 2i-1

    2. Matrix in which the diagonal is "a" and all the rest is "b"

    a , b, ... , b
    b , a,
    .
    .
    .
    b , ... , a


    3. Matrix in which the diagonal is "0". Above the diagonal is "-1" and "1" is below the diagonal. (Need to separate to 2 cases when n is odd and when n is even)

    0,-1,...,-1
    1,0
    .
    .
    .
    1,... 0



    4.

    1,2,...,n
    2,3,...,n+1
    .
    .
    .
    n,n+1,...,2n-1
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Also sprach Zarathustra View Post
    Need to calculate these matrix determinants without using any calculation programs, only by definition!

    Thanks a lot to whom accept this challenge!


    1. Above C


    0 , 3i , 1
    2-i , 3 , 1
    -i , 0 , 2i-1
    Just expand on, say, the first row:
    \left|\begin{array}{ccc}0 & 3i & 1 \\ 2-i & 3 & 1 \\ -i & 0 & 2i-1\end{array}\right|= -3i\begin\left|{array}{cc}2-i & 1 \\ -i & 2i-1\end{array}\right|+ \left|\begin{array}{cc}2- i & 3 \\ -i & 0\end{array}\right|


    For the rest, I recommend looking at the 2 by 2 and 3 by 3 versions and trying to see a pattern.
    2. Matrix in which the diagonal is "a" and all the rest is "b"

    a , b, ... , b
    b , a,
    .
    .
    .
    b , ... , a

    3. Matrix in which the diagonal is "0". Above the diagonal is "-1" and "1" is below the diagonal. (Need to separate to 2 cases when n is odd and when n is even)

    0,-1,...,-1
    1,0
    .
    .
    .
    1,... 0



    4.

    1,2,...,n
    2,3,...,n+1
    .
    .
    .
    n,n+1,...,2n-1
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