Question #135363
Let G be the set of all 3×3 non-singular matrices with entries from Q. Let X be a fixed matrix in G.prove that operation *defined by A*B= X^(-1)ABX be a binary operation. How do you prove that G is a group with respect to binary operation defined by A*B=X^(-1) ABX?
1
Expert's answer
2020-09-28T19:29:51-0400

G={A=(a11a12a13a21a22a23a31a32a33),aijQ,i=1,2,3,j=1,2,3,detA0},X=(x11x12x13x21x22x23x31x32x33),detX0G=\{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

X-fixed matrix in G

detX0X1,detX10detX\neq0\to\exists X^{-1}, detX^{-1}\neq0

X1=1detX(y11y12y13y21y22y23y31y32y33)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}

where yijy_{ij} - algebraic addition to the element xijx_{ij} .

BG:B=(b11b12b13b21b22b23b31b32b33),bijQ,detB0B\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

AB=X1ABXA*B=X^{-1}ABX

X1,A,B,XX^{-1},A,B,X 3 ×\times 3 matrices

ABA*B 3×\times 3 matrices with elemets in Q

det(AB)=det(X1ABX)==det(X1)det(A)det(B)det(X)0ABGdet(A*B)=det(X^{-1}ABX)=\\ =det(X^{-1})det(A)det(B)det(X)\neq0\\ A*B\in G

So * a binary operation.

Prove that G is a group.

1. ABGA*B\in G

2.(AB)C=A(BC)(A*B)*C=A*(B*C)

3.I=(100010001)G,1Q,det(I)=10I=\begin{pmatrix} 1& 0&0 \\ 0& 1&0\\ 0&0&1 \end{pmatrix}\in G,1\in Q, det(I)=1\neq0

4.AA1=IA*A^{-1}=I

Find A1:A^{-1}:

AA1=X1AA1X=X1IX=X1X=Idet(A1)0A*A^{-1}=X^{-1}AA^{-1}X=X^{-1}IX=X^{-1}X=I\\ det(A^{-1})\neq0

So G is a group with respect to binary operation


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