Cogito
AP Precalculus · Chapter 8 · Lesson 3
Matrices as Transformations
A grid of numbers that moves the plane.
12 problems · about 22 minutes · AP Precalculus 4.10, 4.12, 4.13
What this lesson teaches
The student multiplies matrices, computes determinants, and interprets a matrix as a transformation.
- A matrix stores a transformation, applied by multiplication.
- The determinant is the area scaling factor.
- A zero determinant collapses the plane and leaves no inverse.
Warm Up
Straightforward practice. Get the method working first.
5 problems[[4, 1], [3, 2]]. What is the determinant?
Answer 5
Why 5.
What does a determinant of zero mean?
Answer The transformation flattens the plane, so it cannot be undone.
Why No inverse exists.
[[5, 2], [3, 4]]. Determinant?
Answer 14
Why 20 − 6.
[[1, 0], [0, 1]]. Determinant?
Answer 1
Why The identity.
[[6, 3], [4, 2]]. Determinant?
Answer 0
Why 12 − 12.
Build It Up
The same ideas with more to keep track of.
3 problemsDoes that matrix have an inverse? 1 yes, 0 no.
Answer 0
Why A zero determinant.
Determinant 4 applied to a shape of area 7. New area?
Answer 28
Why 7 × 4.
Row (1, 2) against the vector (5, 7). What is the result?
Answer 19
Why 5 + 14.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each determinant value with what it means.
Answer Determinant 1 → Areas are unchanged; Determinant 3 → Areas are tripled; Determinant 0 → The plane collapses onto a line
Why The determinant is an area scaling factor.
[[7, 3], [2, 1]]. Determinant?
Answer 1
Why 7 − 6.
The Determinant: [[3, 1], [2, 4]]. What is the determinant?
Answer 10
Why 10.
The Area: A shape of area 5 under a transformation with determinant 10. What is the new area?
Answer 50
Why 50.