One Number Decides Everything

You'll learnto see the determinant as the number that says whether a transformation can be undone.

The determinant ad minus bc answers two questions at once: whether a matrix can be undone, and how much it scales area. When it is nonzero the matrix has an inverse, built by swapping the diagonal, negating the rest, and dividing. When it is zero the plane has been crushed onto a line and nothing can bring it back. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · DETERMINANT · INVERTIBILITY · AREA · v3

Unit 4 is not assessed on the AP exam; schools include it where local requirements ask.

the shaded region is the unit square, before and after

Before you start

What you're looking at

The unit square at the origin, about to be pushed into a new shape by a matrix.

The question

Can this matrix's transformation be reversed, and how would you tell?

Step 1 / 12