1.9. THE MATHEMATICAL THEORY OF DETERMINANTS 47

Definition 1.9.16 Let A = (ai j) be an n× n matrix. Then a new matrix called thecofactor matrix cof(A) is defined by cof(A) = (ci j) where to obtain ci j delete the ith rowand the jth column of A, take the determinant of the (n−1)× (n−1) matrix which results,(This is called the i jth minor of A. ) and then multiply this number by (−1)i+ j. To makethe formulas easier to remember, cof(A)i j will denote the i jth entry of the cofactor matrix.

The following is the main result. Earlier this was given as a definition and the outra-geous totally unjustified assertion was made that the same number would be obtained byexpanding the determinant along any row or column. The following theorem proves thisassertion.

Theorem 1.9.17 Let A be an n×n matrix where n≥ 2. Then

det(A) =n

∑j=1

ai j cof(A)i j =n

∑i=1

ai j cof(A)i j . (1.25)

The first formula consists of expanding the determinant along the ith row and the secondexpands the determinant along the jth column.

Proof: Let (ai1, · · · ,ain) be the ith row of A. Let B j be the matrix obtained from A byleaving every row the same except the ith row which in B j equals (0, · · · ,0,ai j,0, · · · ,0) .Then by Corollary 1.9.9,

det(A) =n

∑j=1

det(B j)

For example if

A =

 a b cd e fh i j

and i = 2, then

B1 =

 a b cd 0 0h i j

 ,B2 =

 a b c0 e 0h i j

 ,B3 =

 a b c0 0 fh i j

Denote by Ai j the (n−1)× (n−1) matrix obtained by deleting the ith row and the jth

column of A. Thus cof(A)i j ≡ (−1)i+ j det(Ai j). At this point, recall that from Proposition

1.9.6, when two rows or two columns in a matrix M, are switched, this results in multiplyingthe determinant of the old matrix by−1 to get the determinant of the new matrix. Therefore,by Lemma 1.9.15,

det(B j) = (−1)n− j (−1)n−i det((

Ai j ∗0 ai j

))= (−1)i+ j det

((Ai j ∗0 ai j

))= ai j cof(A)i j .

Therefore,

det(A) =n

∑j=1

ai j cof(A)i j