Kenneth Kuttler

80 CHAPTER 5. SOME IMPORTANT LINEAR ALGEBRA

is true for n−1, n≥ 3.∣∣∣∣∣∣∣∣∣∣

1a1+b1

1a1+b2

· · · 1a1+bn

...... · · ·

...1

an−1+b11

an−1+b21

an−1+bn1

an+b11

an+b2· · · 1

an+bn

∣∣∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣∣∣

an−a1(a1+b1)(b1+an)

an−a1(a1+b2)(b2+an)

· · · an−a1(a1+bn)(an+bn)

...... · · ·

...an−an−1

(an−1+b1)(an+b1)an−an−1

(b2+an)(b2+an−1)an−an−1

(an+bn)(bn+an−1)1

an+b11

an+b2· · · 1

an+bn

∣∣∣∣∣∣∣∣∣∣Continuing to use the multilinear properties of determinants, this equals∣∣∣∣∣∣∣∣∣∣

1(a1+b1)(b1+an)

1(a1+b2)(b2+an)

· · · 1(a1+bn)(an+bn)

...... · · ·

...1

(an−1+b1)(an+b1)1

(b2+an)(b2+an−1)1

(an+bn)(bn+an−1)1

an+b11

an+b2· · · 1

an+bn

∣∣∣∣∣∣∣∣∣∣n−1

∏k=1

(an−ak)

and this equals ∣∣∣∣∣∣∣∣∣∣

1(a1+b1)

1(a1+b2)

· · · 1(a1+bn)

...... · · ·

...1

(an−1+b1)1

(b2+an−1)1

(bn+an−1)

1 1 · · · 1

∣∣∣∣∣∣∣∣∣∣∏

n−1k=1 (an−ak)

∏nk=1 (an +bk)

Now take −1 times the last column and add to each previous column. Thus it equals∣∣∣∣∣∣∣∣∣∣

bn−b1(a1+b1)(a1+bn)

bn−b2(a1+b2)(a1+bn)

· · · 1(a1+bn)

...... · · ·

...bn−b1

(b1+an−1)(bn+an−1)bn−b2

(b2+an−1)(bn+an−1)1

(an−1+bn)

0 0 · · · 1

∣∣∣∣∣∣∣∣∣∣∏

n−1k=1 (an−ak)

∏nk=1 (an +bk)

Now continue simplifying using the multilinear property of the determinant.∣∣∣∣∣∣∣∣∣∣

1(a1+b1)

1(a1+b2)

· · · 1...

... · · ·...

1(b1+an−1)

1(b2+an−1)

10 0 · · · 1

∣∣∣∣∣∣∣∣∣∣∏

n−1k=1 (an−ak)

∏nk=1 (an +bk)

∏n−1k=1 (bn−bk)

∏n−1k=1 (ak +bn)

Now, expanding along the bottom row, what has just resulted is∣∣∣∣∣∣∣∣1

a1+b1· · · 1

a1+bn−1... · · ·

...1

an−1+b1· · · 1

an−1+bn−1

∣∣∣∣∣∣∣∣∏

n−1k=1 (an−ak)

∏nk=1 (an +bk)

∏n−1k=1 (bn−bk)

∏n−1k=1 (ak +bn)

80 CHAPTER 5. SOME IMPORTANT LINEAR ALGEBRAis true forn—1,n> 3.1 1 cee 1a,t+b; a, +b2 aytbn1 1 1nV FP ny ¥h2 nt tbnan+b, an+b2 ~ an+bnan—a\ an—4\ wee an-4\(a) +b )(b1+4n) (a) +b2)(b2+an) (a) +bn) (Gn +bn)= Gn—4n-1 Gn—4n-1 Gn—4n-1(Qn—1 +) (an+b1) (b2+an)(bo+an—1) (an+bn)(bn+an—1)ant) Gn+b7 _ Gnt+bnContinuing to use the multilinear properties of determinants, this equals1 1 1(a, +b1)(b1 +4n) (a, +b2)(b2+4an) “— (a1 +bn)(an+bn). . : n—1; ; (Gn ~ ak)(an—1 +b lan +b 1 ) (b2+an)(bo+an—1 ) (an-+bn)(bntan—t y |k=1ant+b; ant+bz ~ an+hnand this equals1 1 I(a1 +61) (ai¥b2) (Gi Fbn)DE | TT Cn = a8)1 1 ee ne a +b(naithi) (atann) Trae) TIi=1 (4n + Bx)1 1 eeNow take —1 times the last column and add to each previous column. Thus it equalsbn—by bn—bo . 1(a +61) (a1 +bn) (a, +b2)(a1+bn) (a) +bn)TIE} (an — ax)bn—by bn—bo 1 TT) (Qn +(6) +4n—1)(bn+an—-1) (b2+4n—1)(bn+an—1) (Gn—1+bn) kel ( " )1Now continue simplifying using the multilinear property of the determinant.I l |(a; +61) (a, +b2); wt | ED @n = ae) TE} (On = Be)1 1 ne b n—1(b) +an—1) (bo +4y—1) 1 Ti (an + X) Mei (ax + bn)0 0 |Now, expanding along the bottom row, what has just resulted is1 1a,+b i ayt+by— _ _. a hot (an — ak) Mot (bn — bx); - ; Tai (an +x) TZ} (ae + bn)Gy—1+By ag FB