Doing one transformation and then another is itself a transformation, and its matrix is the product of the two.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Order changes the answer
Rotating then stretching is not the same as stretching then rotating. That is exactly why AB is not BA.
Determinants multiply
The determinant of AB is the determinant of A times that of B. Two area factors applied in turn compound.
The inverse transformation
The inverse matrix is the transformation that puts everything back. Applying both leaves the plane untouched.
Its determinant
If A scales areas by 5, its inverse scales them by 1/5. Determinants of inverses are reciprocals.
Undoing a composition
The inverse of AB is B inverse times A inverse. Undoing happens in reverse order, like taking off shoes and socks.
Composition is a product
Applying B first and then A is the single transformation given by the product AB. Building complicated transformations from simple ones and multiplying their matrices is how graphics pipelines are constructed.
Right to left
In AB, B acts first. The order in the product is the reverse of the order in time, which follows from how function composition is written and catches people out repeatedly.
Undoing a stack
The inverse of a composition undoes each step in reverse order. That is why (AB)⁻¹ = B⁻¹A⁻¹, and the geometric reading makes the formula obvious rather than something to recall.
Some transformations cannot be undone
A projection collapses a dimension, and the collapsed information is gone. No matrix can restore it, which is exactly what a zero determinant records.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(0, 1)
- j-hat lands on(-1, 0)
- Determinant1
- v lands on(-1, 3)
The shaded parallelogram is the image of the unit square, and its area is 1. That is exactly what the determinant measures.
Step 3: Watch an Example
One step at a time.
Watch Meera Compound Two Factors
Meera applies a transformation of determinant 3 followed by one of determinant 4.
- Step 1
She notes the first transformation triples every area.
Step 4: Your Turn
Practice makes it stick.
The Compound
Problem 1 of 2
Determinants 5 and 2. What is the determinant of the composition?
The Undo
Problem 2 of 2
A has determinant 4. What is the determinant of its inverse, as a decimal?
Stack and Unstack
1 of 8
Determinants 3 and 6. What is the determinant of the composition?
2 of 8
A has determinant 10. What is the determinant of its inverse, as a decimal?
3 of 8
A composition including a matrix of determinant 0. What is the total determinant?
4 of 8
Two reflections, each of determinant −1. What is the determinant of the composition?
5 of 8
The inverse of AB is B inverse times A inverse. Does the order reverse? 1 yes, 0 no.
6 of 8
A transformation followed by its own inverse. What is the total determinant?
7 of 8
Order what happens when the product ABC acts on a vector.
- 1B acts on that result
- 2A acts on that result
- 3The final vector emerges
- 4C acts on the vector
8 of 8
Determinants 2 and −3. What is the determinant of the composition?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Determinants 7 and 2. What is the determinant of the composition?
Question 2 of 2
Why is the inverse of AB equal to B inverse times A inverse?
What You Learned
- 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.