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Math Help - Cofactors, adj, 2x2 matrix

  1. #1
    Member mybrohshi5's Avatar
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    Cofactors, adj, 2x2 matrix

    I really need some clarification about cofactors and adj of a 2x2 matrix

    say the matrix is

    a b
    c d

    what would the cofactors be?

    what would the adj be?

    this is what i think but for some reason i think its different.

    cofactor
    d -c
    -b a


    adj
    d -b
    -c a


    thanks for any clarification about this. i would really appreciate it cause my professor decided to skip this part of the chapter yet its on my online hw

    Thank you
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  2. #2
    Newbie Nissplus's Avatar
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    Colorado USA
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    You had the right answer!

    But as explanation, lets say A is your original matrix:

     A = \begin{array}{lll} a\ \  b \\ c\ \  d \end{array}

    The definitions are best explained backwards:
    . the adjunct adj(A) equals the transpose of the cofactor matrix C
    . the transpose of a matrix is one with the rows and columns flipped
    . The cofactor matrix C is the matrix of minors M, with each position multiplied by its respective sign -1^{r+c}
    . For a 2 x 2 matrix the minor of each position M[r,c] is the diagonally opposite element.

    Using these definitions and starting with A, first create the matrix of minors:

     M = \begin{array}{lll} d\ \ c \\ b\ \ a \end{array}

    Then the cofactor matrix C has each element of M multiplied by its sign -1^{r+c} where r,c are the row and column number of each position.

     C = \begin{array}{lll} \ 1 \times d\ \ -1 \times c \\<br />
-1 \times b\ \ \ 1 \times a \end{array} = \begin{array}{lll} \ d\ \ -c\\ -b\ \ \ a \end{array}

    Finally  adj(A) = C^T so:

     adj(A) = C^T = \begin{array}{lll} \ d\ \ -b\\ -c\ \ \ \ a \end{array}

    The calculation of adj for 3x3 and larger matrices is the same except that the calculation of the minor matrix requires calculating determinants.
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  3. #3
    Member mybrohshi5's Avatar
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    Thank you very much. that was a great explanation
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