Matrices as linear maps

What is it about?

A $2 \times 2$ matrix maps $\vec{x}$ to $A\vec{x}$ . Its columns are the images of $\vec{e}_1$ and $\vec{e}_2$ , so the grid becomes a grid of parallelograms. Applying $B$ first and then $A$ gives the product $A \cdot B$ ; in general $AB \neq BA$ .

$\det A = ad - bc$ is the signed area of the image of the unit square: areas are multiplied by $|\det A|$ , a negative sign reverses orientation, and $\det A = 0$ squashes the plane onto a line (not invertible). Otherwise $A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ .

Eigenvectors satisfy $A\vec{v} = \lambda\vec{v}$ ; the eigenvalues solve $\lambda^2 - (a + d)\lambda + (ad - bc) = 0$ . Rotations by $\varphi \neq 0°, 180°$ have non-real eigenvalues. For an iteration $\vec{x}_{n+1} = A\vec{x}_n$ the eigenvalue of largest absolute value decides the long-term behaviour; for a transition matrix the fixed vector with sum 1 is the stationary distribution.

Try it

General: drag the tips of $A\vec{e}_1$ and $A\vec{e}_2$ ; press “Play the transformation”.
Rotation, reflection in y = x, shear, projection.
AB ≠ BA, eigenvectors, inverse.
Everyday: kaleidoscope, shadow, population, customers switching.

Tasks

Task 1: Matrix from images

$\vec{e}_1 \mapsto (2, 1)$ , $\vec{e}_2 \mapsto (1, 2)$ . Find $A$ , the area factor and the eigenvalues. Load task

Show solution

$A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ , $\det A = 3$ ; $\lambda^2 - 4\lambda + 3 = 0$ gives eigenvalues $3$ and $1$ with eigenvectors $(1, 1)$ and $(1, -1)$ . Show

Task 2: Order matters

Compute $AB$ and $BA$ for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ , $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ . Load task

Show solution

$AB = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$ (columns swapped), $BA = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$ (rows swapped). Show

Task 3: Inverse

Find the inverse of $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ . Load task

Show solution

$\det A = 1$ , so $A^{-1} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$ . Show

Task 4: Stationary distribution

Find the stationary distribution of $\begin{pmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{pmatrix}$ . Load task

Show solution

$0.9p + 0.2q = p$ gives $p = 2q$ ; with $p + q = 1$ : $p = \frac{2}{3}$ , $q = \frac{1}{3}$ . Show

Ideas for the classroom

Let students build rotations, reflections and shears by dragging, then write down the matrices and compare with the standard forms.
Switch “Show values” off: compute determinant, product, inverse or eigenvalues first, then check in the picture.
Common stumbling blocks: swapping rows and columns; thinking $A$ acts first in $AB$ ; ignoring the sign of the determinant; believing eigenvectors stay unchanged (they may be stretched or flipped).