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How to find C - the cofactor matrix of A?

  1. Feb 26, 2012 #1
    1. The problem statement, all variables and given/known data
    I need a way to find C - the cofactor matrix of A, assuming that A can be any arbitary matrix.


    2. Relevant equations
    See 1.


    3. The attempt at a solution
    Tried Googling without much success.
     
  2. jcsd
  3. Feb 26, 2012 #2
    Do you know what the cofactor matrix actually is?
     
  4. Feb 26, 2012 #3

    HallsofIvy

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    What do you mean by "find the cofactor matrix"? Are you looking for a formula so you can just plug in the values of a? Such a thing can be found but it would be terribly complicated. The simplest way to find it is to use the definition- each value, [itex]C_{mn}[/itex], is the mn-cofactor of A; that is, the determinant of the matrix you get by removing the mth row and nth column of A.
     
  5. Feb 26, 2012 #4
    I was looking for a fomula to find a cofactor matrix of any other matrix. The reason why, is because I want to be able to inverse any matrices, without only using rules of thumb which work for simple 2x2 and 2x3 matrices.

    As the inverse of a matrix A is 1/|A|*adj(A), and adj(A) is the transpose of a cofactor matrix C of A, I need it to find adjugates of any matrix, to later apply it to inverse matrices.
     
  6. Feb 26, 2012 #5
    I'll try and explain it step by step, please correct me if I'm wrong.

    Lets take a random matrix,

    [tex]\begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix}[/tex]

    Then you take the cofactor of each element.

    To find the cofactor take the element, eg. [itex]A_{11}[/itex] then you delete the row and column that the elment is in. Then you find the determinant of the resultant matrix.

    So for our matrix,

    [itex]A_{11}[/itex], so delete row 1 and column 1.

    This leaves us with,

    [tex]\begin{bmatrix} 4 & 5 \\ 0 & 6 \end{bmatrix}[/tex]

    Then take the determinant of that matrix.

    [tex]\begin{vmatrix} 4 & 5 \\ 0 & 6 \end{vmatrix}[/tex] = ad-bc = 4 * 6 - 5 * 0 = 24.

    So the first element, [itex]A_{11}[/itex], of the cofactor matrix is 24.

    [tex]\begin{bmatrix}24 & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{bmatrix}[/tex]

    Just repeat the steps for the rest of the elements.
     
  7. Feb 26, 2012 #6
    Thanks a lot, this helped.
     
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