5.16. Expansion Formulas by Cofactors

Every row A i =( a i1 ,, a in )  of an n ´ n matrix A can be expressed as a linear combination of the n unit vectors I 1 ,, I n , i.e.,

        A i = j=1 n a ij I j

Since determinants are linear in each row, we have

        d( , A i , )=d( , j=1 n a ij I j , )   = j=1 n a ij d( , I j , )             (5.19)

We now introduce A ij  as the matrix obtained from A by replacing its ith row with the unit vector Ij.  For example, for n=3 , we have

        A 11 =( 1 0 0 a 21 a 22 a 23 a 31 a 32 a 33 )   A 12 =( 0 1 0 a 21 a 22 a 23 a 31 a 32 a 33 )    A 13 =( 0 0 1 a 21 a 22 a 23 a 31 a 32 a 33 )

Thus

det A ij =d( A 1 ,, I j i ,, A n )  

so that (5.19) can be written as

        detA= j=1 n a ij det A ij                          (5.23)

where i can be any integer between 1 and n.  The quantity det A ij  is call the cofactor of the element a ij  and is denoted by

cof   a ij =det A ij                                        (5.22)

so that (5.20) becomes Theorem 5.15:

        detA= j=1 n a ij cof   a ij                          (5.24)

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, det E α =+1  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.