Build 2×2 matrices and read their determinant in the sandbox, and predict — before you read it off the screen — how much each one scales area, whether it flips orientation, and the determinant of a composition of two of them.
The determinant of a 2×2 matrix is one number that tells the whole story of what it does to area: |det| is the factor the unit square's area is multiplied by, a negative sign means orientation is flipped, and det = 0 means the plane has collapsed onto a line — no inverse exists. And because det(AB) = det(A)·det(B), composing transformations just multiplies their determinants.
Open the sandbox. Each step names the 2×2 matrix to set from its four entries a b c d; type them into the matrix inputs or drag the column handles, and read the determinant and area readouts.
Matrix A -> [ a b ; c d ] (set the four entries each step names)
Read: det(A) and signed area of the image of the unit square
Watch the determinant readout and the shaded image of the unit square in the main panel — every number you predict is printed there.
Set A = [2, 0; 0, 3] — a pure scaling. Its determinant is ad - bc. Predict det(A), then read the determinant readout.
The readout shows det(A) = ad - bc = 2·3 - 0·0 = 6: the transformation multiplies every area by 6, so the unit square maps to a 2×3 rectangle.
Set A = [-1, 0; 0, 1] — a reflection across the y-axis. Predict the signed area of the image of the unit square, then read it.
The signed area of the image is -1: the magnitude |det| = 1 means area is preserved, but the negative sign means orientation is flipped — the square is turned over like a mirror image.
Compose A = [1, 1; 0, 1] (a shear, det = 1) with B = [2, 0; 0, 3] (det = 6). Predict det(AB), then read it.
The readout shows det(AB) = 6, exactly det(A)·det(B) = 1·6. The shear adds no area, so composing it with B leaves B's area factor unchanged.
Everything above is waiting in the sandbox. Drag the column handles to build any 2×2 matrix, watch the unit square and the illustrative shape transform in real time, and see the determinant, the eigenvectors and the composition update at once.
- Read a scaling matrix's determinant, det = ad - bc = 6, as an area factor
- Saw a reflection's signed area come out -1 — area preserved, orientation flipped
- Confirmed the product law det(AB) = det(A)·det(B) = 6
- Every value you predicted is the demo's own matrix arithmetic, not a picture