What you already know, renamed
In Volume I you learned that some matrices have an inverse and some do not, and that an n-by-n matrix is invertible exactly when its determinant is nonzero. Volume II asks a deeper question: forget any single matrix — what is the structure of the whole collection of invertible matrices? The answer is a group, and that one shift in viewpoint opens the entire theory of continuous symmetry.
A group is a set with one operation that (1) is closed — combining two members gives a member, (2) is associative, (3) has an identity element, and (4) gives every member an inverse inside the set. For matrices the operation is matrix multiplication. Check the four axioms and you discover the collection was a group all along.
The general linear group GL(n)
The general linear group GL(n) is the set of all invertible n-by-n real matrices under multiplication. It is closed because det(AB) = det(A)det(B), so a product of nonzero-determinant matrices again has nonzero determinant. The identity is I, and each member's inverse is its matrix inverse. Unlike addition, this group is non-commutative for n >= 2 — AB and BA usually differ.
# GL(2): two invertible matrices, both with det != 0 A = [2, 1; 1, 1] det(A) = 2*1 - 1*1 = 1 (invertible) B = [1, 1; 0, 1] det(B) = 1*1 - 1*0 = 1 (invertible) A*B = [2, 3; 1, 2] det = 4 - 3 = 1 <- closed: still in GL(2) B*A = [3, 2; 1, 1] det = 3 - 2 = 1 <- but B*A != A*B (non-abelian) A^-1 = [1, -1; -1, 2] A * A^-1 = [1, 0; 0, 1] = I
Carving out SL(n) by a condition
Most interesting groups are subgroups carved out of GL(n) by an extra condition. The special linear group SL(n) keeps only the matrices with determinant exactly 1. Why is this a group? Because det(AB) = det(A)det(B) = 1*1 = 1 keeps you inside, and det(A^-1) = 1/det(A) = 1 too. Geometrically SL(n) is the group of volume-preserving, orientation-preserving transformations.
Why these groups are continuous
GL(n) is not a finite checkerboard of symmetries — you can slide one matrix smoothly into another by nudging its entries while keeping the determinant nonzero. A group that is also a smooth surface, where multiplication and inversion are continuous, is a Lie group. That marriage of algebra (the group law) and geometry (a smooth manifold) is the subject of everything that follows.
Counting dimensions makes this concrete. GL(n) lives in the n-by-n matrices, so it has n^2 free parameters. SL(n) imposes one equation, det = 1, dropping the dimension to n^2 - 1. Symmetry groups become shapes you can measure, and the rest of this arc measures them.