Question

o b au nvertiie matrices in MaCR) under the Aroup cr r bu

Answer 1

Let

メ メ

to prove that H is a subgroup of the group G of all invertible matrices in M₂(R) under multiplication[M₂(R) means 2 by 2 matrices with real entries]

[to prove that H is a subgroup we have to prove

(i)identity belong to H

(ii) H is closed under multiplication ; that is if $A\in H,B\in H\Rightarrow AB\in H$

(iii) inverse exist in H]

proof

(i) the identity matrix

  
01
since
1メ0

so identity element exist in H

(ii) to prove H is closed under matrix multiplication

let

:a, bメ0

$B=\begin{bmatrix} c &0 \\ 0&d \end{bmatrix};c,d\neq 0$

АЕН, В Е Н

$AB=\begin{bmatrix} a &0 \\ 0& b \end{bmatrix}\begin{bmatrix} c &0 \\ 0& d \end{bmatrix}$

  $=\begin{bmatrix} ac &0 \\ 0& bd \end{bmatrix}$

$AB\in H$$ac\neq 0;bd\neq 0$   $;ac\in R,bd\in R$

$A\in H,B\in H\Rightarrow AB\in H$

so H is closed under multiplication

(iii) to prove the existence of inverse in H

$A=\begin{bmatrix} a &0 \\ 0& b \end{bmatrix}$

det(A)=ab

$A^{-1}=\frac{1}{ab}\begin{bmatrix} b &0 \\ 0& a \end{bmatrix}\in H$ [since and also belongs to R]

2 1a

so inverse exists for each element of H

hence H is a subgroup of G of all invertible matrices in M₂(R) under multiplication