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8.6. A FORMULA FOR THE INVERSE 179

Theorem 8.5.3 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 . (8.11)

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 8.3.2, det(A) = ∑

nj=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

8.2.3, 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 8.5.1,

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) = ∑nj=1 ai j cof(A)i j which is the formula for expanding det(A) along the

ith row. Also,

det(A) = det(AT )= n

∑j=1

aTi j cof

(AT )

i j =n

∑j=1

a ji cof(A) ji

which is the formula for expanding det(A) along the ith column. ■

8.6 A Formula for the InverseNote that this gives an easy way to write a formula for the inverse of an n×n matrix.

Theorem 8.6.1 A−1 exists if and only if det(A) ̸= 0. If det(A) ̸= 0, then the i jth entry ofA−1 is given by a−1

i j where a−1i j = det(A)−1 cof(A) ji for cof(A)i j the i jth cofactor of A.

8.6. A FORMULA FOR THE INVERSE 179Theorem 8.5.3 Let A be ann X n matrix where n > 2. Thenn ndet(A) = )° aijcof(A);; = )° aijcof(A);;. (8.11)j=l i=lThe first formula consists of expanding the determinant along the i‘ row and the secondexpands the determinant along the j'" column.Proof: Let (aj1,-+- ,din) be the i” row of A. Let B '; be the matrix obtained from A byleaving every row the same except the i” row which in Bj equals (0,--- ,0,a;;,0,--- ,0).Then by Corollary 8.3.2, det (A) = )_, det (Bj) . For example ifabeA=|de fh i jand i = 2, thenabe abe abeBi=| d 0 0 ],Bo=] 0 e€ 0 |,B=] 0 0 fFhii j hi j hi jDenote by A‘! the (n—1) x (n—1) matrix obtained by deleting the i” row and the j’”column of A. Thus cof (A);; = (—1)'*Y det (A’/) . At this point, recall that from Proposition8.2.3, 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 8.5.1,nj (_1)"-i de Au xdet(Bj)) = (—1)"4(-1) oe (( 0 a)naci(( “ “ )) = ajjcof(A);;.Therefore, det (A) = )}_ aij cof(A);; which is the formula for expanding det (A) along thejthrow. Also,det (A) = det (A’) = Ynajjcot (4°), = Ye ajicof (A) ;j=l j=lwhich is the formula for expanding det (A) along the i” column. ll8.6 A Formula for the InverseNote that this gives an easy way to write a formula for the inverse of an n x n matrix.Theorem 8.6.1 A! exists if and only if det(A) 40. If det(A) 4 0, then the ij!" entry ofA”! is given by qj," where a;;' = det(A)! cof (A) ;; for cof (A);; the ij" cofactor of A.