The Determinant as Area

One number that says how much space was stretched.

12 problems · about 23 minutes · N-VM.C.12

What this lesson teaches

The student computes 2 by 2 determinants and interprets them as signed area factors.

The determinant ad − bc is the factor by which a transformation scales every area.
Its sign records whether the plane was flipped over.
A determinant of zero means everything collapsed onto a line, so no inverse can exist.

Straightforward practice. Get the method working first.

5 problems

Rows [6, 2] and [4, 3]. What is the determinant?
Answer 10
Why 10.
What does the determinant measure?
Answer The factor by which the transformation multiplies areas.
Why An area scale factor.
Rows [4, 1] and [2, 3]. What is the determinant?
Answer 10
Why 12 − 2.
Rows [2, 6] and [1, 3]. What is the determinant?
Answer 0
Why 6 − 6.
Rows [5, 0] and [0, 5]. What is the determinant?
Answer 25
Why Multiply the diagonal.

The same ideas with more to keep track of.

3 problems

A determinant of 0. What is the area of the image of the unit square?
Answer 0
Why It has collapsed.
A determinant of 7 on a shape of area 2. What is the new area?
Answer 14
Why Multiply.
Rows [1, 2] and [3, 4]. What is the determinant?
Answer -2
Why 4 − 6.

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

4 problems

Sort each determinant by what it says about the transformation.
Answer Keeps orientation: Determinant 3, Determinant 1 · Flips or collapses: Determinant −3, Determinant 0
Why Only positive determinants keep orientation.
The identity matrix. What is its determinant?
Answer 1
Why Nothing is stretched.
The Stretch: A determinant of 6 applied to a shape of area 3. What is the new area?
Answer 18
Why 18.
The Flip: A determinant of −4 applied to a shape of area 5. What is the new area?
Answer 20
Why 20.