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Math Help - Determinants/Adjugates, etc.

  1. #1
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    Determinants/Adjugates, etc.

    I have a couple questions...

    I don't understand when you're supposed to put a negative sign infront of the matrix when you switch two rows/columns of the matrix. When we first started matrices (reducing them,) we didn't do this, but now that we're taking determinants we do. I'm just not sure when I'm supposed to place a negative sign in front and when not to. (I'm talking about when two rows/columns of the matrix are switched.)

    Also, when finding the determinant of a matrix, we can reduce the matrix by removing a factor of a row/column and placing it on the outside of the matrix. This must be done for EACH row/column when using determinants, but when we first did it, we could remove one factor and it would effect the WHOLE matrix, not just one row/column.

    Any explanations?
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Thomas View Post
    I have a couple questions...

    I don't understand when you're supposed to put a negative sign infront of the matrix when you switch two rows/columns of the matrix. When we first started matrices (reducing them,) we didn't do this, but now that we're taking determinants we do. I'm just not sure when I'm supposed to place a negative sign in front and when not to. (I'm talking about when two rows/columns of the matrix are switched.)

    Also, when finding the determinant of a matrix, we can reduce the matrix by removing a factor of a row/column and placing it on the outside of the matrix. This must be done for EACH row/column when using determinants, but when we first did it, we could remove one factor and it would effect the WHOLE matrix, not just one row/column.

    Any explanations?

    for the first one, i could answer it this way..
    let A_1, ..., A_i be the column of a matrix.
    det([A_1 ... A_k ... A_j .... A_i]) = (-1)^p det([A_1 ... A_j ... A_k .... A_i]) where p= j-k..

    the second, i just cant figure out what your question is.. can you give an example?
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  3. #3
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    I understand how to do it, I don't understand WHEN. For example, when we reduced matrices to reduced row echelon form, we never had to place a negative sign in front of the matrix when switching two rows. BUT, no we are supposed to do determinants. So when I'm using a matrix now, I'm not sure if I'm supposed to place a negative sign or not anymore.

    My second question.. hmm... think about a 3x3 matrix which consists of three's on the first row, three's on the second row, and three's on the third row. In order to reduce this matrix, we have to take out a a factor of 3^3, which makes 27 on the outside of the matrix and reduces the matrix to all one's. BUT, BEFORE all we would of done was take out a single 3, and this would of reduced the WHOLE matrix to one's.

    Do you understand why questions?
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  4. #4
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Thomas View Post
    I understand how to do it, I don't understand WHEN. For example, when we reduced matrices to reduced row echelon form, we never had to place a negative sign in front of the matrix when switching two rows. BUT, no we are supposed to do determinants. So when I'm using a matrix now, I'm not sure if I'm supposed to place a negative sign or not anymore.
    that actually answers your own question. take note that there are only 3 row operations, and you use these row operations when we reduce matrices. also take note that determinant is actually a function which maps a matrix to a scalar. so based on the arguments, a REMARK would be: you will do the "negation" when you are dealing with determinants. (hmm, maybe if you would see the proof of the "negative" thing, you would understand more.

    Quote Originally Posted by Thomas View Post
    My second question.. hmm... think about a 3x3 matrix which consists of three's on the first row, three's on the second row, and three's on the third row. In order to reduce this matrix, we have to take out a a factor of 3^3, which makes 27 on the outside of the matrix and reduces the matrix to all one's. BUT, BEFORE all we would of done was take out a single 3, and this would of reduced the WHOLE matrix to one's.

    Do you understand why questions?
    the second question maybe a consequence of the first
    hmm, maybe you are mixing matrices and determinants too much.
    i'm really sorry i cannot explain it well, i hope others could explain it for me..
    Last edited by kalagota; October 24th 2007 at 04:52 AM.
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