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Math Help - a question on 4x4 matrix

  1. #1
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    a question on 4x4 matrix

    A and B are real 4x4 matrices, if det(B)=1 and B=2A^3(A)T (T means transpose), what is the det(A)? Thank you very much in advance for your time and effort.
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  2. #2
    Member eXist's Avatar
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    From how I interpreted your problem:

     det(B) = 1
     B = 2A^3A^T

    If that's the case, remember determinants can be broken up over multiplication.

    So if I re-write B:

     B = 2A^3A^T = 2 A A A A^T

    Now that you have that, be careful when you multiply by a scalar (in this case 2). Think about how that changes the matrix A in terms of row operations.
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  3. #3
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    Hi eXist,

    Thanks very much for your reply. Could you please explain further as I am a newbie to matrix. from det(B)=1, I can see B is an identity matrix and det(2AAA(A)T)=1. so how can I solve det(A)? Do I need to find the matrix A? Thank you and please excuse me for my poor knowledge on this.

    cheers,
    pingping
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  4. #4
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    Quote Originally Posted by shenpingping View Post
    Hi eXist,

    Thanks very much for your reply. Could you please explain further as I am a newbie to matrix. from det(B)=1, I can see B is an identity matrix and det(2AAA(A)T)=1. so how can I solve det(A)? Do I need to find the matrix A? Thank you and please excuse me for my poor knowledge on this.

    cheers,
    pingping
    From what you have been told you should see that:

    2^n [\det(A)]^3\det(A^T)=1

    also you should know that \det(A^T)=\det(A) so:

    2^n [\det(A)]^4=1

    where n is the size of the matrix in this case n=4

    (when you multiply a matrix by a scalar every element is multiplied by the scalar, and for an n \times n matrix the determinant can be written as a sum of \pm the product of n terms at a time from the matrix, this is where the 2^n term above comes from)

    CB
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  5. #5
    Member eXist's Avatar
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    Quote Originally Posted by CaptainBlack View Post

    2^n [\det(A)]^4=1

    where n is the size of the matrix in this case n=4
    Yep, CB beat me to it. Just wanted you to see that multiplying a matrix by a scalar, changes the determinant of the matrix by that scalar to the nth power. n being the size of the matrix.

    Thanks CB.
    -Chad
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  6. #6
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    Thumbs up Thank you so much

    Hi CaptainBlack,

    Thanks you very much for your explanation. greatly appreciate.

    cheers,
    pingping
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