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Cogito

Linear Algebra · Chapter 5 · Lesson 1

Practice Worksheet

The Determinant as Area

One number that says how much space was stretched.

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

NameDate

Figure — use these to answer the problems

The shaded parallelogram is the image of the unit square, and its area is the determinant. Drive it to zero and watch it collapse.
ij
  • i-hat lands on(3, 1)
  • j-hat lands on(1, 2)
  • Determinant5
a3
b1
c1
d2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Rows [6, 2] and [4, 3]. What is the determinant?

    Answer

  2. What does the determinant measure?

    • The factor by which the transformation multiplies areas.
    • The sum of the entries.

  3. Rows [4, 1] and [2, 3]. What is the determinant?

    Answer

  4. Rows [2, 6] and [1, 3]. What is the determinant?

    Answer

  5. Rows [5, 0] and [0, 5]. What is the determinant?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A determinant of 0. What is the area of the image of the unit square?

    Answer

  2. A determinant of 7 on a shape of area 2. What is the new area?

    Answer

  3. Rows [1, 2] and [3, 4]. What is the determinant?

    Answer

Stretch Yourself

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

4 problems

  1. Sort each determinant by what it says about the transformation.

    Write each item under the heading it belongs to: Determinant 3 · Determinant −3 · Determinant 1 · Determinant 0

    Keeps orientation

    Flips or collapses

  2. The identity matrix. What is its determinant?

    Answer

  3. The Stretch

    A determinant of 6 applied to a shape of area 3. What is the new area?

    Answer

  4. The Flip

    A determinant of −4 applied to a shape of area 5. What is the new area?

    Answer