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.

8 STEPS · 6 QUICK CHECKS · DETERMINANT · INVERTIBILITY · AREA · v2

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

the shaded region is the unit square being transformed

Before you start

What you're looking at

Two grids side by side: one still two-dimensional, one crushed onto a line.

The question

Can this matrix's transformation be reversed?

Watch for

Zero versus nonzero determinant deciding invertibility before the formula ad − bc appears.

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