Thread: Determinant from Cofactor

Generally, a determinant A is defined as e_ijk..na_1ia_2ja_3k.. Then a co-factor A_rs is defined and it is shown (arduously) that A=a_LiA_Li. Or, a determinant is defined inductiveley in terms of defined A_rs, 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=e_LMNe_ijka_Lia_Mja_Nk

A=a_Li{(-1)^(L+i)e_MNe_jka_Mja_Nk}^*

A_Li=(-1)^(L+i)e_MNe_jka_Mja_Nk

A=a_LiA_Li

a_QiA_Li=e_LMNe_ijka_Qia_Mja_Nk=0 if Q unequal L because then Q=M or N

End


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

^*Proof of e_LMNe_ijk=(-1)^(L+i)e_MNe_jk:
Suppose abc..L.. are in sequential order and L is in rth place. Then
e_Labc.... = (-1)^(r-1)e_abc…
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
e_LMN=(-1)^(L-1)e_MN, and similarly e_ijk=(-1)^(i-1)e_jk so that e_LMNe_ijk=(-1)^(L+i)e_MNe_jk

Note: For general case of cofactor, replace LMN with LMN,…, and ijk wiith ijk,…, in derivation.