In the realm of geometry, we have the powerful tools of matrices to effortlessly manipulate points, lines, shapes, and curves through reflection, rotation, and translation transformations. Each of these transformations corresponds to a distinct matrix, making geometric manipulation both systematic and elegant.

Transforming points, lines, shapes and curves using reflection, rotation and translations. Define a matrix for each transformation. How can we use an inverse matrix to find the original point, given its image? The determinant of a transformation matrix is associated with the scaling factor of the area. By taking the absolute value of the determinant, you can determine how the transformation affects the area of a shape. This concept becomes invaluable when analyzing the change in shape or size resulting from transformations. Certain points and lines remain fixed or unchanged under specific transformations. These are known as invariant points and lines. For instance, the centre of rotation remains invariant during a rotation. Recognizing these invariants can simplify geometric analysis.