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Lemma 2.2. Let R be a ring, and M ∈ M n (R). Let M ij denote the (i, j)-minor of M , 1 ≤ i, j ≤ n. We have the following two identities.

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2 Preliminary

In this section, we shall give some useful lemmas and formulas. These lemmas and formulas will be used in next section.

Lemma 2.1. Let k be a field, and G ≤ GL n (k) be a finite group and let G act on k[x 1 , · · · , x n ] and k(x 1 , · · · , x n ). Let a ∈ k[x 1 , · · · , x n ] and f (T ) be the minimal polyno- mial of a over k(x 1 , · · · , x n ) G . Then f (T ) ∈ k[x 1 , · · · , x n ] G [T ].

Next, we give two useful formulas.

Lemma 2.2. Let R be a ring, and M ∈ M n (R). Let M ij denote the (i, j)-minor of M , 1 ≤ i, j ≤ n. We have the following two identities.

(a) Let N ∈ M n −2 (R) be the matrix obtained by deleting the first and the last rows, the first and the last columns from M . Then

detM · detN = M nn M 1,1 − M n,1 M 1,n .

(b) Let N 0 ∈ M n −2 (R) be the matrix obtained by deleting the last two rows and the first and the last columns from M . Then

detM · detN 0 = M nn M n −1,1 − M n1 M n −1,n .

Before stating Lemma 2.3, we give some notations.

Let R be any commutative ring and A = (a ij ) ∈ M n (R). Let I = {1, 2, · · · , n}.

For S = {s 1 , · · · , s r } and T = {t 1 , · · · , t r } ⊂ I, we denote A ST to be the r-minor obtained from the s 1 , · · · , s r rows and the t 1 , · · · , t r columns of A. We denote S 0 to be the complement I \ S. We also define l(S, T ) = s 1 + · · · + s r + t 1 + · · · + t r .

Lemma 2.3. Let A = (a ij ), X = (x ij ) ∈ M n (R). Then the determinant of A + X is

|a ij + x ij | = X n

r=0

{ X

|S|=|T |=r

( −1) l(S,T ) A ST X S

0

T

0

}.

The proofs of Lemma 2.1, Lemma 2.2 and Lemma 2.3 can be found in [8].

3

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