Signed area
The Reflect preset $\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}$ has $\det = 1 \cdot (-1) - 0 = -1$. Area cannot be negative — so what is the minus sign recording?
Under Identity, Scale, Shear and Rotate 30°, the arc from A to B always sweeps counterclockwise and $\det A$ stays positive. Reflect leaves A in place but sends B below the axis: the sweep now runs clockwise, the stats read $\det A = -1.00$, area $\times\, 1.00$. Reflect + Scale reads $\det A = -1.50$ — flipped ordering, areas multiplied by $1.50$.
The determinant is a signed area. Its size, $|\det A|$, is lesson 1's area factor. Its sign records orientation: whether the map preserves the counterclockwise ordering of the square's corners or reverses it. A reflection is a flip — no stretching, but every loop that ran counterclockwise now runs clockwise, so $\det = -1$. Composing the flip with a scale multiplies the size without unflipping, giving $-1.5$.
The sign is also why flips are special: no amount of turning or stretching crosses from $+$ to $-$ without passing through $0$. Watch $\det A$ during the animation into Reflect — it slides down through zero on the way to $-1$.
About this model
What it represents: the sign of a $2\times2$ determinant as the orientation of the ordered pair of columns — positive when the A→B sweep stays counterclockwise, negative when it reverses. For the plane, this picture is exact.
What it leaves out: the general definition. In $n$ dimensions the sign tracks the orientation of an ordered basis and is defined through the determinant's algebraic properties — multilinearity and the sign flip on swapping columns — not through a visible corner arc.
Source: Margalit and Rabinoff, Interactive Linear Algebra, Determinants: Definition (Georgia Tech, free online) — the defining properties that carry the sign to $n \times n$.
Reflect + Scale has $\det A = -1.5$. A patch of area $2$ goes through the map. What comes out?
Which of these matrices reverses orientation?
Transfer: run Reflect, then Reflect again. What is the determinant of the combined motion?
Read $\det A$ in two parts: $|\det A|$ is the area factor, and the sign is an orientation bit — positive keeps the corner ordering, negative flips it. The sign can only change by passing through zero, and lesson 3 is about exactly that crossing.