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