Determinants and Their Applications in Mathematical Physics

3.7 Bordered Determinants 47

and letBndenote the determinant of order (n+ 1) obtained by bordering

Anby the column

X=

[

x 1 x 2 x 3 ···xn

]T

on the right, the row

Y=

[

y 1 y 2 y 3 ···yn

]

at the bottom and the elementzin position (n+1,n+ 1). In some detail,

Bn=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a 11 a 12 ··· a 1 n x 1

a 21 a 22 ··· a 2 n x 2

........................

an 1 an 2 ··· ann xn

y 1 y 2 ··· yn z

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

. (3.7.1)

Some authors border on the left and at the top but this method displaces

the elementaijto the position (i+1,j+ 1), which is undesirable for both

practical and aesthetic reasons except in a few special cases.

In the theorems which follow, the notation is simplified by discarding the

suffixn.

Theorem 3.9.

B=zA−

n

∑

r=1

n

∑

s=1

Arsxrys.

Proof. The coefficient ofysinBis (−1)

n+s+1

F, where

F=

∣

∣C

1 ...Cs− 1 Cs+1...CnX

∣

∣

n

=(−1)

n+s

G,

where

G=

∣

∣

C 1 ...Cs− 1 XCs+1...Cn

∣

∣

n

.

The coefficient ofxrinGisArs. Hence, the coefficient ofxrysinBis

(−1)

n+s+1+n+s

Ars=−Ars.

The only term independent of the x’s and y’s is zA. The theorem

follows.

LetEijdenote the determinant obtained fromAby

a.replacingaijbyz,i, jfixed,

b.replacingarjbyxr,1≤r≤n,r=i,

c. replacingaisbyys,1≤s≤n,s=j.

Theorem 3.10.

Bij=zAij−

n

∑

r=1

n

∑

s=1

Air,jsxrys=Eij.