Question

Let \(G\) be a subgroup of \(GL_2(\mathbb{R})\) generated by 

Then the order of \(G\) is ...........

Hint:

The order of a subgroup generated by two matrices can be found by calculating the distinct products of powers of the matrices and checking their closure under matrix multiplication.

Answer 1

Correct Answer: 5.9

Step 1: Understanding the given group. 
We are given a subgroup \(G\) of \(GL_2(\mathbb{R})\) generated by two matrices: 

The order of the subgroup is the number of distinct elements that can be formed by taking all possible products of powers of these matrices. 

Step 2: Checking the properties of the matrices. 
- \(A\) is an involution (i.e., \(A^2 = I\), the identity matrix). - We compute the powers of \(B\) and their products with \(A\) to check the order of the subgroup. Through calculation, it is determined that the group formed by \(A\) and \(B\) has 4 distinct elements. This is the minimal order of the group generated by these two matrices. 
Step 3: Conclusion. 
The order of \(G\) is 4.