G = { A = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , a i j ∈ Q , i = 1 , 2 , 3 , j = 1 , 2 , 3 , d e t A ≠ 0 } , X = ( x 11 x 12 x 13 x 21 x 22 x 23 x 31 x 32 x 33 ) , d e t X ≠ 0 G=\{A=\begin{pmatrix}
a_{11} & a_{12}&a_{13} \\
a_{21} & a_{22}&a_{23} \\
a_{31} & a_{32}&a_{33}
\end{pmatrix}, a_{ij}\in Q,\\
i=1,2,3,j=1,2,3, detA\neq0 \},\\
X=\begin{pmatrix}
x_{11} & x_{12}&x_{13} \\
x_{21} & x_{22}&x_{23} \\
x_{31} & x_{32}&x_{33}
\end{pmatrix}, detX\neq0 G = { A = ⎝ ⎛ a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 ⎠ ⎞ , a ij ∈ Q , i = 1 , 2 , 3 , j = 1 , 2 , 3 , d e t A = 0 } , X = ⎝ ⎛ x 11 x 21 x 31 x 12 x 22 x 32 x 13 x 23 x 33 ⎠ ⎞ , d e tX = 0
X-fixed matrix in G
d e t X ≠ 0 → ∃ X − 1 , d e t X − 1 ≠ 0 detX\neq0\to\exists X^{-1}, detX^{-1}\neq0 d e tX = 0 → ∃ X − 1 , d e t X − 1 = 0
X − 1 = 1 d e t X ( y 11 y 12 y 13 y 21 y 22 y 23 y 31 y 32 y 33 ) X^{-1}=\frac{1}{detX}\begin{pmatrix}
y_{11} & y_{12}&y_{13} \\
y_{21} & y_{22}&y_{23} \\
y_{31} & y_{32}&y_{33}
\end{pmatrix} X − 1 = d e tX 1 ⎝ ⎛ y 11 y 21 y 31 y 12 y 22 y 32 y 13 y 23 y 33 ⎠ ⎞
where y i j y_{ij} y ij - algebraic addition to the element x i j x_{ij} x ij .
B ∈ G : B = ( b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ) , b i j ∈ Q , d e t B ≠ 0 B\in G: B=\begin{pmatrix}
b_{11} & b_{12}&b_{13} \\
b_{21} & b_{22}&b_{23} \\
b_{31} & b_{32}&b_{33}
\end{pmatrix}, b_{ij}\in Q,\\
detB\neq0 B ∈ G : B = ⎝ ⎛ b 11 b 21 b 31 b 12 b 22 b 32 b 13 b 23 b 33 ⎠ ⎞ , b ij ∈ Q , d e tB = 0
A ∗ B = X − 1 A B X A*B=X^{-1}ABX A ∗ B = X − 1 A BX
X − 1 , A , B , X X^{-1},A,B,X X − 1 , A , B , X 3 × \times × 3 matrices
A ∗ B A*B A ∗ B 3× \times × 3 matrices with elemets in Q
d e t ( A ∗ B ) = d e t ( X − 1 A B X ) = = d e t ( X − 1 ) d e t ( A ) d e t ( B ) d e t ( X ) ≠ 0 A ∗ B ∈ G det(A*B)=det(X^{-1}ABX)=\\
=det(X^{-1})det(A)det(B)det(X)\neq0\\
A*B\in G d e t ( A ∗ B ) = d e t ( X − 1 A BX ) = = d e t ( X − 1 ) d e t ( A ) d e t ( B ) d e t ( X ) = 0 A ∗ B ∈ G
So ∗ * ∗ a binary operation.
Prove that G is a group.
1. A ∗ B ∈ G A*B\in G A ∗ B ∈ G
2.( A ∗ B ) ∗ C = A ∗ ( B ∗ C ) (A*B)*C=A*(B*C) ( A ∗ B ) ∗ C = A ∗ ( B ∗ C )
3.I = ( 1 0 0 0 1 0 0 0 1 ) ∈ G , 1 ∈ Q , d e t ( I ) = 1 ≠ 0 I=\begin{pmatrix}
1& 0&0 \\
0& 1&0\\
0&0&1
\end{pmatrix}\in G,1\in Q, det(I)=1\neq0 I = ⎝ ⎛ 1 0 0 0 1 0 0 0 1 ⎠ ⎞ ∈ G , 1 ∈ Q , d e t ( I ) = 1 = 0
4.A ∗ A − 1 = I A*A^{-1}=I A ∗ A − 1 = I
Find A − 1 : A^{-1}: A − 1 :
A ∗ A − 1 = X − 1 A A − 1 X = X − 1 I X = X − 1 X = I d e t ( A − 1 ) ≠ 0 A*A^{-1}=X^{-1}AA^{-1}X=X^{-1}IX=X^{-1}X=I\\
det(A^{-1})\neq0 A ∗ A − 1 = X − 1 A A − 1 X = X − 1 I X = X − 1 X = I d e t ( A − 1 ) = 0
So G is a group with respect to binary operation
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