Results 1 to 1 of 1

Math Help - Determinant from Cofactor

  1. #1
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Determinant from Cofactor

    Generally, a determinant A is defined as eijk..na1ia2ja3k.. Then a co-factor Ars is defined and it is shown (arduously) that A=aLiALi. Or, a determinant is defined inductiveley in terms of defined Ars, in which case lABl = lAllBl is stated without proof, proved quite arduously, or, in one case, assigned as an exercise!!

    (convention- sum over lower case letters)

    Determinant and Derived Cofactor:

    A=eLMNeijkaLiaMjaNk

    A=aLi{(-1)(L+i)eMNejkaMjaNk}*

    ALi=(-1)(L+i)eMNejkaMjaNk

    A=aLiALi

    aQiALi=eLMNeijkaQiaMjaNk=0 if Q unequal L because then Q=M or N

    End


    Definition:
    eabc…=1 with a,b,c,.. any positive integers in sequential order.
    eabc…= +1 or -1 according as abc..are an even or odd permutation of the sequential order.

    *Proof of eLMNeijk=(-1)(L+i)eMNejk:
    Suppose abc..L.. are in sequential order and L is in rth place. Then
    eLabc.... = (-1)(r-1)eabc…
    If now abc.. are permuted, both sides still hold.
    If LMN are put in sequential order, L will be in Lth place and then in general
    eLMN=(-1)(L-1)eMN, and similarly eijk=(-1)(i-1)ejk so that eLMNeijk=(-1)(L+i)eMNejk

    Note: For general case of cofactor, replace LMN with LMN,…, and ijk wiith ijk,…, in derivation.
    Last edited by Hartlw; October 1st 2012 at 04:43 PM. Reason: forgot to add sum convention
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: December 14th 2010, 12:45 AM
  2. Cofactor of a Determinant
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 4th 2010, 09:41 AM
  3. Replies: 1
    Last Post: November 20th 2009, 09:14 AM
  4. Matrix doing a cofactor expansion
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 2nd 2009, 05:10 PM
  5. Cofactor Expansion
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 20th 2008, 01:14 PM

Search Tags


/mathhelpforum @mathhelpforum