Final answer:
A matrix with a determinant of 1 preserves volume during transformation, is invertible, and is related to orthogonal matrices, which indicates that it maintains certain distance relationships between points.
Step-by-step explanation:
When a matrix has a determinant of 1, it has several important implications. One critical aspect is that it represents a linear transformation that preserves volume in the space it transforms. For instance, in two dimensions, a 2x2 matrix with a determinant of 1 would transform a shape without changing its area. In three dimensions, a 3x3 matrix with a determinant of 1 would transform a solid without altering its volume.
Additionally, a determinant of 1 indicates that the matrix is invertible, meaning there exists another matrix that can reverse the transformation effected by the original matrix. This inverse matrix also has a determinant of 1. Furthermore, the matrix with a determinant of 1 also preserves distance relationships between points to a certain extent, which is associated with the concept of it being an orthogonal matrix.