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Math Help - Cramer's Rule and Determinants

  1. #1
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    Cramer's Rule and Determinants

    I am to solve the following using Cramer's rule:

    -x1 - 4x2 + 2x3 + x4 = -32
    2x1 - x2 + 7x3 + 9x4 = 14
    -x1 + x2 + 3x3 + x4 = 11
    x1 - 2x2 + x3 - 4x4 = -4

    I know x1 = [det(A1)]/[det(A)]

    but I'm not sure what the best approach is to find the determinant of a 4x4 matrix, because cofactor expansion takes a very long time, would reducing the matrix to a triangular matrix be the best approach?

    Thanks
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    Quote Originally Posted by Adrian View Post
    I am to solve the following using Cramer's rule:

    -x1 - 4x2 + 2x3 + x4 = -32
    2x1 - x2 + 7x3 + 9x4 = 14
    -x1 + x2 + 3x3 + x4 = 11
    x1 - 2x2 + x3 - 4x4 = -4

    I know x1 = [det(A1)]/[det(A)]

    but I'm not sure what the best approach is to find the determinant of a 4x4 matrix, because cofactor expansion takes a very long time, would reducing the matrix to a triangular matrix be the best approach?

    Thanks
    i think it would be almost the same amount of work. it may be a little, just a little less work to transform the matrices into triangular matrices. do as you wish, if you want to use cofactor expansion method, expand along the first column
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  3. #3
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    So for cofactor expansion I should find the detA to start off, but I'm not sure how to find the determinant of a 4x4 matrix?

    -1 -4 2 1
    2 -1 7 9
    -1 1 3 1
    1 -2 1-4

    Do I find the determinants of each of the 3x3 matrices I am left with after going through each of the numbers in column 1?

    If so, I should start at the -1 in Row 1 Column 1
    That leaves me with:
    -1 7 9
    1 3 1
    -2 1 -4
    and the determinant is -1(-12-2) + -7(-4+2) + 9(1+6) = 91

    Then move on to the 2nd number in column 1 of the 4x4 matrix?
    That leaves me with:
    -4 2 1
    1 3 1
    -2 1 -4
    And find the determinant of that.

    Then do the same thing with the -1 in Row 2 and the 1 in Row 4 of column 1 of the 4x4 matrix.

    If I am correct so far, after going through all 4 numbers in Column 1 would I add the results of their determinants together? And this would be my determinant of the 4x4 matrix?

    Thanks, but if I can learn to find the determinant of a 4x4 matrix then I should be able to deal with this problem.
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    Quote Originally Posted by Adrian View Post
    So for cofactor expansion I should find the detA to start off, but I'm not sure how to find the determinant of a 4x4 matrix?

    -1 -4 2 1
    2 -1 7 9
    -1 1 3 1
    1 -2 1-4

    Do I find the determinants of each of the 3x3 matrices I am left with after going through each of the numbers in column 1?

    If so, I should start at the -1 in Row 1 Column 1
    That leaves me with:
    -1 7 9
    1 3 1
    -2 1 -4
    and the determinant is -1(-12-2) + -7(-4+2) + 9(1+6) = 91

    Then move on to the 2nd number in column 1 of the 4x4 matrix?
    That leaves me with:
    -4 2 1
    1 3 1
    -2 1 -4
    And find the determinant of that.

    Then do the same thing with the -1 in Row 2 and the 1 in Row 4 of column 1 of the 4x4 matrix.

    If I am correct so far, after going through all 4 numbers in Column 1 would I add the results of their determinants together? And this would be my determinant of the 4x4 matrix?

    Thanks, but if I can learn to find the determinant of a 4x4 matrix then I should be able to deal with this problem.
    if what you're saying is this, then you are correct.

    \left| \begin{array}{cccc} -1 & -4 & 2&1 \\ 2& -1& 7& 9 \\ -1 &1 &3 &1 \\ 1& -2& 1&-4 \end{array} \right| = - \left| \begin{array}{ccc}  -1&7 &9 \\ 1& 3& 1\\ -2& 1&-4 \end{array} \right| - 2 \left| \begin{array}{ccc}  -4& 2& 1 \\ 1& 3& 1\\ -2& 1&-4 \end{array} \right| - \left| \begin{array}{ccc}  -4& 2& 1\\ -1& 7& 9\\ -2& 1&-4 \end{array} \right| - \left| \begin{array}{ccc}  -4& 2& 1\\ -1& 7& 9\\ 1& 3& 1\end{array} \right|
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    And inside each of those 3x3 matrices I have to find the determinant correct?

    Yes that is what I was saying except I forgot the scalars in front of each 3x3 matrix, although may I ask why the scalars 2 and 1 are negative in your solution. They are the 5th and 13th numbers in the 4x4 matrix meaning they should both be positive and not negative shouldn't it? I thought it went like this for a 4x4 matrix
    + - + -
    + - + -
    + - + -
    + - + -


    I know 3x3 matrices go like this:
    + - +
    - + -
    + - +
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Adrian View Post
    And inside each of those 3x3 matrices I have to find the determinant correct?

    Yes that is what I was saying except I forgot the scalars in front of each 3x3 matrix, although may I ask why the scalars 2 and the 1 are negative in your solution. They are the 5th and 13th numbers in the 4x4 matrix meaning they should both be positive and not negative shouldn't it? I thought it went like this for a 4x4 matrix
    + - + -
    + - + -
    + - + -
    + - + -
    no, you always use the checkerboard pattern, starting with a + at the top left, so the sign structure for 4x4 matrices are:

    + - + -
    - + - +
    + - + -
    - + - +

    I know 3x3 matrices go like this:
    + - +
    - + -
    + - +
    yes
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  7. #7
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    Alright, that makes sense. Thanks a lot, looks like I'll be able to finish this problem now, although finding the determinant of 4 various 4x4 matrices for Cramer's Rule should take quite some time.

    Thanks again
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Adrian View Post
    Alright, that makes sense. Thanks a lot, looks like I'll be able to finish this problem now, although finding the determinant of 4 various 4x4 matrices for Cramer's Rule should take quite some time.

    Thanks again
    indeed, it will take long. good luck
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