 When "ad" − "bc" ≠ 0, "q" is an invertible matrix, and then

The collection of all such invertible matrices constitutes the general linear group GL(2, R).In terms of abstract algebra, M(2, R) with the associated addition and multiplication operations forms a ring, and GL(2, R) is its group of units.M(2, R) is also a four-dimensional vector space, so it is considered an associative algebra.