Every row of an n ´ n matrix A can be expressed as a linear combination of the n unit vectors , i.e.,
Since determinants are linear in each row, we have
We now introduce as the matrix obtained from A by replacing its ith row with the unit vector Ij. For example, for , we have
so that (5.19) can be written as
where i can be any integer between 1 and n. The quantity is call the cofactor of the element and is denoted by
so that (5.20) becomes Theorem 5.15:
Obviously, one can apply (5.23,4) to the cofactors repeatedly to reduce detA to a linear combination entirely of determinants of matrices Ea that are related to the unit matrix I by purely row permutations. By Theorem 5.3, or -1 for even or odd permutations, respectively. Thus, eqs(5.23,4) are expansion formulae for evaluating detA. More specifically, they are expansions by the ith row cofactors.