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Matrix singular problem

  1. May 13, 2009 #1
    1. The problem statement, all variables and given/known data
    Let A be a non-singular n x n matrix with a non-zero cofactor Ann and let

    c = det(A) / Ann

    Show that if we subtract c from ann, then the resulting matrix will be singular.


    2. Relevant equations

    det(A) = a1nA1n+...+annAnn


    3. The attempt at a solution

    Well, if I replace det(A) with the one in "Relevant eq.", and multiply both sides with Ann I get:

    cAnn = a1nA1n+...+annAnn

    Then if I subtract cAnn from both sides i get:

    0 = a1nA1n+...+annAnn - cAnn
    , which we can rewrite to:

    0 = a1nA1n+...+(ann-c)Ann


    And now I'm not sure if I'm done ?

    It seems like I need to define another matrix of some sort to define det(B) = 0.
    But I'm not quite sure how I do that. Can anyone give me a hint ? :)
     
  2. jcsd
  3. May 13, 2009 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yes, define a matrix B which is exactly the same as A except for the (n,n) element, which will be

    [tex]a_{nn} - c[/tex] instead of [tex]a_{nn}[/tex]

    Then calculate det(B) the same way you calculated det(A), and compare the answers you get.
     
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