Practice question · Multiple choice
A 2x2 matrix is invertible exactly when ad − bc ≠ 0. Why should a single number decide whether a transformation can be undone?
Hints
- The determinant measures area scaling. What does scaling area by zero do to a shape?
- If two different points get sent to the same place, can the map be undone?
Show the answer
C. Because a zero determinant collapses the plane onto a line or point.
Why
The determinant is the area scale factor, so zero means the unit square's image has no area, the plane has been flattened onto a line. Infinitely many points then land on one, and an inverse would have to choose between them with nothing to go on. The formula divides by the determinant because of this, not the other way round.
Practise Determinants
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