Given an n ´ n matrix , the n ´ n matrix
is called the cofactor matrix of A.
For any n ´ n matrix , we have
where t is the transpose operation so that . Thus, if , then A-1 exists and is given by
Using Theorem 5.15, we expand detA by its kth row cofactors as
Let B be a matrix equal to A except for the ith ( ) row, which equals to the kth row of A. Thus, so that by Theorem 5.3(b), . Expanding detB by its ith row cofactors gives
and for all j
which is often called the expansion by alien cofactors. Eqs(5.27,9) can be combined to give
This proves (5.25) since
A square matrix A is nonsingular iff .
This is simply the corollary of Theorems 5.12 and 5.16.