Composition and the Inverse Transformation

Stacking transformations, and undoing the stack.

12 problems · about 23 minutes · N-VM.C.8, N-VM.C.10

NameDate

Figure — use these to answer the problems

Set this to a rotation, then to a stretch, and note where v lands each way. The two orders give different answers.

b-1

v, first entry3

v, second entry1

Figure 1

Straightforward practice. Get the method working first.

5 problems

Determinants 7 and 2. What is the determinant of the composition?
Answer
2
Why is the inverse of AB equal to B inverse times A inverse?
- Undoing a sequence means undoing the last step first.
- The order makes no difference.
Determinants 3 and 6. What is the determinant of the composition?
Answer
A has determinant 10. What is the determinant of its inverse, as a decimal?
Answer
A composition including a matrix of determinant 0. What is the total determinant?
Answer

The same ideas with more to keep track of.

3 problems

Two reflections, each of determinant −1. What is the determinant of the composition?
Answer
The inverse of AB is B inverse times A inverse. Does the order reverse? 1 yes, 0 no.
Answer
A transformation followed by its own inverse. What is the total determinant?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Order what happens when the product ABC acts on a vector.
Write 1 to 4 in the boxes to put these in order.
- C acts on the vector
- B acts on that result
- A acts on that result
- The final vector emerges
Determinants 2 and −3. What is the determinant of the composition?
Answer
The Compound
Determinants 5 and 2. What is the determinant of the composition?
Answer
The Undo
A has determinant 4. What is the determinant of its inverse, as a decimal?
Answer