determin.ant — Linear Algebra from Scratch

Where you'll see this: volume calculations in physics simulations, change-of-variables in integration (the Jacobian determinant), checking whether a set of vectors can form a coordinate system, and computing cross products in 3D graphics — all use the determinant as a measure of "how much space is being scaled."

Every linear transformation scales areas by some fixed factor. Double the matrix, double the area. Rotate — area is unchanged. Collapse to a line — area goes to zero. That scaling factor is the determinant.

The unit square

The unit square has corners at (0,0), (1,0), (0,1), and (1,1). Its area is exactly 1. After a linear transformation, those four corners move to new positions. The parallelogram they form has area equal to $|\det(A)|$ .

The formula

For a 2×2 matrix, the determinant has a compact formula:

\det\begin{bmatrix}a & b \\ c & d\end{bmatrix} = ad - bc

This looks arbitrary until you think geometrically. $ad$ is the area of the rectangle spanned by the diagonal entries. $bc$ is the area "lost" due to the off-diagonal shearing. The difference is the actual parallelogram area.

Sign encodes orientation

The determinant can be negative. A negative value means the transformation flips orientation — what was counter-clockwise becomes clockwise, like a mirror reflection. The magnitude still gives the area scaling; the sign tells you whether space was flipped.

Try the Flip preset. The determinant is −1: area is preserved (magnitude 1) but the grid flips. Try Shear — the parallelogram leans over but the area stays surprisingly constant.

Scaling the whole matrix

If you multiply every entry by a scalar $s$ , the determinant scales by $s^2$ for a 2×2 matrix:

\det(sA) = s^2 \det(A)

Makes sense: scale both dimensions by $s$ , and area scales by $s \times s = s^2$ .

Area Scaling

The unit square

The formula

Sign encodes orientation

Scaling the whole matrix