5.17. The Cofactor Matrix

Definition:  Cofactor Matrix.

Given an n ´ n matrix A=( a ij ) , the n ´ n matrix

        cofA= ( cof a ij ) i,j=1 n

is called the cofactor matrix of A.

Theorem 5.16.

For any n ´ n matrix A=( a ij ) , we have

        A   ( cofA ) t =( detA )I                               (5.25)

where t is the transpose operation so that ( A t ) ij = a ji .  Thus, if detA0 , then A-1 exists and is given by

        A 1 = 1 detA ( cofA ) t                          (5.26)

Proof

Using Theorem 5.15, we expand detA by its kth row cofactors as

        detA= j=1 n a kj cof a kj                          (5.27)

Let B be a matrix equal to A except for the ith ( ik  ) row, which equals to the kth row of A.  Thus, B i = A k = B k  so that by Theorem 5.3(b), detB=0 .  Expanding detB by its ith row cofactors gives

        detB= j=1 n b ij cof b ij =0                              (5.28)

Using

        b ij = a kj        and  cof b ij =cof a ij             for all j

eq(5.28) becomes

        j=1 n a kj cof a ij =0                 where ki                      (5.29)

which is often called the expansion by alien cofactors.  Eqs(5.27,9) can be combined to give

        j=1 n a kj cof a ij = δ ki detA                              (5.30)

This proves (5.25) since

        j=1 n a kj cof a ij = [ A   ( cofA ) t ] ki

Theorem 5.17

A square matrix A is nonsingular iff detA0 .

Proof

This is simply the corollary of Theorems 5.12 and 5.16.