Kenneth Kuttler

178 CHAPTER 8. DETERMINANTS

=C(n,m)

∑k=1

∑{r1,··· ,rm}=Ik

1m!

sgn(r1 · · ·rm)2 det(Bk)det(Ak) =

C(n,m)

∑k=1

det(Bk)det(Ak)

since there are m! ways of arranging the indices {r1, · · · ,rm}. ■

8.5 Expansion Using CofactorsLemma 8.5.1 Suppose a matrix is of the form

M =

(A ∗0 a

)or

(A 0

∗ a

)(8.10)

where a is a number and A is an (n−1)× (n−1) matrix and ∗ denotes either a columnor a row having length n− 1 and the 0 denotes either a column or a row of length n− 1consisting entirely of zeros. Then det(M) = adet(A) .

Proof: Denote M by (mi j) . Thus in the first case, mnn = a and mni = 0 if i ̸= n while inthe second case, mnn = a and min = 0 if i ̸= n. From the definition of the determinant,

det(M)≡ ∑(k1,··· ,kn)

sgnn (k1, · · · ,kn)m1k1 · · ·mnkn

Letting θ denote the position of n in the ordered list, (k1, · · · ,kn) then using the earlierconventions used to prove Lemma 8.1.1, det(M) equals

∑(k1,··· ,kn)

(−1)n−θ sgnn−1

(k1, · · · ,kθ−1,

kθ+1, · · · ,n−1kn

)m1k1 · · ·mnkn

Now suppose the second case. Then if kn ̸= n, the term involving mnkn in the above expres-sion equals zero. Therefore, the only terms which survive are those for which θ = n or inother words, those for which kn = n. Therefore, the above expression reduces to

a ∑(k1,··· ,kn−1)

sgnn−1 (k1, · · ·kn−1)m1k1 · · ·m(n−1)kn−1 = adet(A) .

To get the assertion in the first case, use Corollary 8.3.1 to write

det(M) = det(MT )= det

((AT 0

∗ a

))= adet

(AT )= adet(A)■

In terms of the theory of determinants, arguably the most important idea is that ofLaplace expansion along a row or a column. This will follow from the above definition ofa determinant.

Definition 8.5.2 Let A = (ai j) be an n× n matrix. Then a new matrix called the cofactormatrix cof(A) is defined by cof(A) = (ci j) where to obtain ci j delete the ith row and thejth column of A, take the determinant of the (n−1)× (n−1) matrix which results, (Thisis called the i jth minor of A. ) and then multiply this number by (−1)i+ j. To make theformulas easier to remember, cof(A)i j will denote the i jth entry of the cofactor matrix.

The following is the main result.

178 CHAPTER 8. DETERMINANTSC(n,m) C(nm)1= y y — sgn (r tee Im)” det (Bx) det (Ax) = y det (By) det (Ax)m!k=l {ry tm}at k=1since there are m! ways of arranging the indices {71,--- , 7m}.8.5 Expansion Using CofactorsLemma 8.5.1 Suppose a matrix is of the formu=(4 2 jor(4 2) (8.10)Oa x awhere a is a number and A is an (n—1) x (n—1) matrix and * denotes either a columnor a row having length n— and the 0 denotes either a column or a row of length n—1consisting entirely of zeros. Then det (M) = adet (A).Proof: Denote M by (mj;) . Thus in the first case, myn = a and my; = 0 if i An while inthe second case, my, = a and mj, = 0 if i~n. From the definition of the determinant,det (M) = y sgn, (ki,°*+ skn) Mk, °° * nk,(k1 ++ kn)Letting @ denote the position of n in the ordered list, (k;,--- ,k,) then using the earlierconventions used to prove Lemma 8.1.1, det (M/) equalsn—-6 @ n—1y (-1) sgn,—1 ky-++ sKo-1,Ko4i5+++ 5 kn Mk, °° nk,(kt skin)Now suppose the second case. Then if k, #n, the term involving m,,,, in the above expres-sion equals zero. Therefore, the only terms which survive are those for which 0 = n or inother words, those for which k,, =n. Therefore, the above expression reduces toa y sgn,—| (k1,-++Kn—-1) Mk, “MN n—-1)ky_1 =adet(A).(kt. skn—1)To get the assertion in the first case, use Corollary 8.3.1 to writeA’ 0* adet (M) = det (M") = det (( )) =adet (A’) = adet(A) iIn terms of the theory of determinants, arguably the most important idea is that ofLaplace expansion along a row or a column. This will follow from the above definition ofa determinant.Definition 8.5.2 Let A = (ajj) be ann xn matrix. Then a new matrix called the cofactormatrix cof (A) is defined by cof(A) = (ci;) where to obtain c;; delete the i" row and thei" column of A, take the determinant of the (n—1) x (n—1) matrix which results, (Thisis called the ij" minor of A. ) and then multiply this number by (—1)'*!. To make theformulas easier to remember, cof (A); j will denote the i j'” entry of the cofactor matrix.The following is the main result.