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.

A powerful shortcut: the columns of a transformation matrix are the images of the standard basis vectors. If a linear transformation sends $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ to $\begin{pmatrix} p \\ q \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ to $\begin{pmatrix} r \\ s \end{pmatrix}$, then $M = \begin{pmatrix} p & r \\ q & s \end{pmatrix}$. This lets you build the matrix of any described transformation directly, and it explains every standard matrix: the rotation matrix has columns $(\cos\theta, \sin\theta)$ and $(-\sin\theta, \cos\theta)$ because those are the images of the unit vectors after turning through $\theta$.