Rotation and reflection matrices

Problem

Let $A$ and $B$ be the rotations about the origin by $90°$ clockwise and counterclockwise, and let $C$ and $D$ be the reflections in the lines $x=y$ and $x=-y$. Pick the correct statements about images of points, negatives of matrices, inverses, and compositions.

The full question and options are on the CEEC 112 GSAT Mathematics A paper.

Where it goes wrong

Hanlin’s solution notes (a publisher’s analysis, not CEEC) rate this item medium-easy; it tests the properties of rotation and reflection matrices. The hard part is not heavy computation. It is getting three things right at once: the signs for clockwise and counterclockwise, the direction of each mirror line, and the rule that the transformation applied first is written on the right of a product. Memorizing only the shape of the matrices often leads to mistaking $C^{-1}$ for the other reflection, or to assuming every pair of transformations commutes.

Idea

The matrix of a counterclockwise rotation by $\theta$ about the origin is

so

The two reflection matrices are

Reflecting twice returns to the start, so $C^{-1}=C$ and $D^{-1}=D$. In a matrix product the transformation applied first is written on the right. Multiplying directly gives

The two rotations commute here; a rotation and a reflection usually do not. Check each pair by its geometric action or by multiplying it out. “Matrix multiplication usually does not commute” does not mean every pair is unequal.

What the figure shows

• Each product plays in two steps. For example, $AC$ applies $C$ first, then $A$.
• Two rotations: compare $AB$ and $BA$, and check that both return to the start.
• Rotation and reflection: compare $AC$ and $CA$, and watch the same two transformations split apart when the order changes.
• AB vs CD: one side is the identity, the other a $180°$ rotation.
• AC vs BD: two different compositions land on the same reflection in the $x$-axis.