Verification of Subgroup Properties for 2x2 Matrices with Determinant One

Suppose we have a set of 2 by 2 matrices whose determinants are non-zero. Consider the subset of these matrices whose determinant is 1. Using the two-step subgroup test, verify if this subset is closed under multiplication and if an element belongs to this subset, confirm that its inverse also belongs to this subset.

In this problem, we explore the properties of a set of two by two matrices with determinant equal to one, commonly known as the special linear group, SL(2, R), where R represents the field of real numbers. The task is to determine whether this set forms a subgroup under matrix multiplication. This involves confirming closure under multiplication and the existence of inverses within this set—a key characteristic of subgroup structure within abstract algebra.

The two-step subgroup test is a standard approach to verify subgroup properties, critical in understanding group structures. The first step confirms closure under the operation, ensuring that multiplying two matrices from the subgroup results in another matrix that remains in the subgroup. The second step involves showing that for every matrix in the subgroup, its inverse is also within the subgroup. For this specific case, since the determinant of the inverse of a matrix is the reciprocal of the determinant, any matrix with determinant one will also have an inverse with determinant one, satisfying the subgroup criteria.

This problem provides an opportunity to apply abstract algebraic concepts such as group operations and inverse elements to matrix groups. It helps build intuition about the implications of determinant properties and their role in maintaining structure within mathematical sets under certain operations.