Quick questions on Matrices and linear transformations of the plane (QCE Specialist Mathematics Unit 3)

A $2\times 2$ matrix multiplies a column vector to produce a new vector. For

The absolute value $|\det M|$ is the factor by which areas are scaled under the transformation: a unit square of area $1$ maps to a parallelogram of area $|\det M|$. The sign of $\det M$ records orientation: a negative determinant means the transformation includes a reflection (orientation is reversed).

To apply transformation $A$ then transformation $B$ to a vector, compute $B(A\mathbf{v}) = (BA)\mathbf{v}$. The combined matrix is the product $BA$, with the second transformation on the left. Order matters: rotating then reflecting generally differs from reflecting then rotating. The determinant of a product equals the product of determinants, $\det(BA) = \det B \, \det A$, so the area-scaling factors multiply.

Reflection is applied second, so it goes on the left:

$\det(FR) = (0)(0) - (-1)(-1) = -1$. This equals $\det F \cdot \det R = (-1)(1) = -1$, confirming the reflection reverses orientation while preserving area.