Cogito
Linear Algebra · Chapter 4 · Lesson 3
Composition and the Inverse Transformation
Stacking transformations, and undoing the stack.
12 problems · about 23 minutes · N-VM.C.8, N-VM.C.10
What this lesson teaches
The student composes transformations and relates the inverse matrix to reversing the motion.
- Composing transformations multiplies their matrices, and the order matters.
- Determinants multiply too, because area factors compound.
- The inverse of a composition undoes the steps in reverse order.
Warm Up
Straightforward practice. Get the method working first.
5 problemsDeterminants 7 and 2. What is the determinant of the composition?
Answer 14
Why 14.
Why is the inverse of AB equal to B inverse times A inverse?
Answer Undoing a sequence means undoing the last step first.
Why Reverse order.
Determinants 3 and 6. What is the determinant of the composition?
Answer 18
Why Multiply.
A has determinant 10. What is the determinant of its inverse, as a decimal?
Answer 0.1
Why One over 10.
A composition including a matrix of determinant 0. What is the total determinant?
Answer 0
Why Anything times zero.
Build It Up
The same ideas with more to keep track of.
3 problemsTwo reflections, each of determinant −1. What is the determinant of the composition?
Answer 1
Why Two flips restore orientation.
The inverse of AB is B inverse times A inverse. Does the order reverse? 1 yes, 0 no.
Answer 1
Why Shoes and socks.
A transformation followed by its own inverse. What is the total determinant?
Answer 1
Why Nothing has changed.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsOrder what happens when the product ABC acts on a vector.
Answer 1. C acts on the vector 2. B acts on that result 3. A acts on that result 4. The final vector emerges
Why Rightmost acts first.
Determinants 2 and −3. What is the determinant of the composition?
Answer -6
Why Keep the sign.
The Compound: Determinants 5 and 2. What is the determinant of the composition?
Answer 10
Why 10.
The Undo: A has determinant 4. What is the determinant of its inverse, as a decimal?
Answer 0.25
Why 0.25.