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Math Help - [SOLVED] matrix determinant

  1. #1
    Senior Member euclid2's Avatar
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    [SOLVED] matrix determinant

    if we have the following matrix
    2 1 0 0
    3 2 0 0
    1 1 3 4
    2 -1 2 3 and we want to calculate the determinant .shouldn't in any correct way of calculation obtain the same value of determinant?
    if we change the 4th column by making c4----- 3c4 -4c3 we will have determinant =3
    if we change the 3rd column by making c3--------- 4c3- 3c2 we will have determinant= 4
    if we change in lines we will have determinant = 1
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  2. #2
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    Quote Originally Posted by euclid2 View Post
    if we have the following matrix
    2 1 0 0
    3 2 0 0
    1 1 3 4
    2 -1 2 3 and we want to calculate the determinant .shouldn't in any correct way of calculation obtain the same value of determinant?
    if we change the 4th column by making c4----- 3c4 -4c3 we will have determinant =3
    if we change the 3rd column by making c3--------- 4c3- 3c2 we will have determinant= 4
    if we change in lines we will have determinant = 1


    If B results from A by interchanging two rows or columns, then det(B)=-det(A)
    If B results from A by multiplying one row or column with the number c, then det(B)=c*det(A)
    If B results from A by adding a multiple of one row to another row, or a multiple of one column to another column, then det(B)=det(A)

    sorry about the first post, apparently i can't copy and paste formulae
    Last edited by Tarzan; February 10th 2009 at 09:56 AM.
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  3. #3
    Member TheMasterMind's Avatar
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    Quote Originally Posted by euclid2 View Post
    if we have the following matrix
    2 1 0 0
    3 2 0 0
    1 1 3 4
    2 -1 2 3 and we want to calculate the determinant .shouldn't in any correct way of calculation obtain the same value of determinant?
    if we change the 4th column by making c4----- 3c4 -4c3 we will have determinant =3
    if we change the 3rd column by making c3--------- 4c3- 3c2 we will have determinant= 4
    if we change in lines we will have determinant = 1
    Consider this:

    (1) we add the product of column 3 by -4 to the column 4

    2 1 0 0
    3 2 0 0
    1 1 3 -8
    2 -1 2 -5

    or

    (2) we add the product of column 2 by -3 to the column 3

    2 1 -3 0
    3 2 -6 0
    1 1 0 4
    2 -1 5 3

    If we multiply a column/row by a factor the determinant is multiplied by the same factor.

    In your two examples the determinant is multiplied by

    (1) 3

    (2) 4

    Another way to compute the determinant is to transform the original matrix into a upper or lower triangular matrix. Starting with

    Matrix A

    2 1 0 0
    3 2 0 0
    1 1 3 4
    2 -1 2 3

    det A=1

    and applying the pivot technique we get the following equivalent Lower Triangular Matrix

    Matrix LTM

    1/2 0 0 0
    3 2 0 0
    -5/3 7/3 1/3 0
    2 -1 2 3

    det LTM = product of the main diagonal entries = (1/2)2(1/3)3 = 1
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  4. #4
    Senior Member euclid2's Avatar
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    Quote Originally Posted by TheMasterMind View Post
    Consider this:

    (1) we add the product of column 3 by -4 to the column 4

    2 1 0 0
    3 2 0 0
    1 1 3 -8
    2 -1 2 -5

    or

    (2) we add the product of column 2 by -3 to the column 3

    2 1 -3 0
    3 2 -6 0
    1 1 0 4
    2 -1 5 3

    If we multiply a column/row by a factor the determinant is multiplied by the same factor.

    In your two examples the determinant is multiplied by

    (1) 3

    (2) 4

    Another way to compute the determinant is to transform the original matrix into a upper or lower triangular matrix. Starting with

    Matrix A

    2 1 0 0
    3 2 0 0
    1 1 3 4
    2 -1 2 3

    det A=1

    and applying the pivot technique we get the following equivalent Lower Triangular Matrix

    Matrix LTM

    1/2 0 0 0
    3 2 0 0
    -5/3 7/3 1/3 0
    2 -1 2 3

    det LTM = product of the main diagonal entries = (1/2)2(1/3)3 = 1
    So the determinant is 1?....Thanks
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  5. #5
    Member TheMasterMind's Avatar
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    Quote Originally Posted by euclid2 View Post
    So the determinant is 1?....Thanks
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