Question

Proove that the set of oil 2x2 matrices with entries from R and determinant +1 is a group under multiplication

Matrix transformations

Proove that the set of oil $$2 \times 2$$ matrices with entries from R and determinant +1 is a group under multiplication

We already know that the set of all invertible $$2 \times 2$$ matrices with entries from R is a group under multiplication — notation: GL(2,R). The given set (of all matrices of determinant 1) is a subset of the set of all invertible matrices. Thus, it is sufficient to prove that this is a subgroup of GL(2,R).
Let A, B be two matrices with determinant 1. Then $$\displaystyle{B}^{{-{{1}}}}$$ exists, and using Binet-Cauchy Theorem,
$$\displaystyle{B}^{{-{{1}}}}={I}\Rightarrow{\det{{\left({B}^{{-{{1}}}}\right)}}}={\det{{\left({I}\right)}}}\Rightarrow{\det{{B}}}\ {{\det{{B}}}^{{-{{1}}}}=}{1}\Rightarrow{\det{{B}}}=^{{-{{1}}}}$$
Therefore, det $$\displaystyle{B}^{{-{{1}}}}={1}$$. This means that $$\displaystyle{\det{{\left({A}{B}^{{-{{1}}}}\right)}}}={\det{{A}}}\ {{\det{{B}}}^{{-{{1}}}}=}{1}\cdot{1}={1}$$
Therefore, $$\displaystyle{A}{B}^{{-{{1}}}}!$$ is a $$2 \times 2$$ matrix with entries from R and determinant 1, which proves that this is a subgroup of GL(2, B), and it is itself a group.