Determinants and What They Measure · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Reading a determinant as a scale factor

Mathematics · Linear Algebra · ages 18-19
Name ______________________   Date ____________

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

    Answer: ______________

  2. What is the determinant of the matrix with rows (3, 1) and (2, 4)?

    Answer: ______________

  3. A map has determinant 2. What does it do to regions?

    • Areas halve and orientation flips
    • Areas double and orientation is kept
    • Areas stay the same

  4. What is the determinant of the matrix with rows (4, 7) and (0, 3)?

    • 7
    • 12
    • 21

  5. A square coefficient matrix has determinant zero. What follows?

    • Its rows are dependent and the system has no unique solution
    • Its rows are independent and the solution is unique
    • Its inverse has determinant zero
    • Its columns span a larger space

  6. What is the determinant of the matrix with rows (1, 2, 3), (0, 1, 4) and (0, 0, 2)?

    Answer: ______________

  7. Vectors (3, 0) and (0, 4) span a parallelogram. What is its area?

    Answer: ______________

  8. What is the determinant of [[4, 7], [0, 3]]?

    • 12
    • 7
    • 21
    • 0

  9. A map has determinant minus 3, and a region of area 5 passes through it. What comes out?

    • Area 15 with orientation flipped
    • Area 8 with orientation kept
    • Area 2 with orientation flipped

  10. What is the determinant of [[1, 2, 3], [0, 1, 4], [0, 0, 2]]?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Reading a determinant as a scale factor W1-mt_fWr2D_1BdM-s1

  1. 10 · Determinant is 3 times 4 minus 1 times 2 = 12 minus 2 = 10.
  2. 10 · Use a times d minus b times c: 3 times 4 minus 1 times 2 is 10.
  3. Areas double and orientation is kept · Size 2 scales area by 2, and a positive sign keeps orientation.
  4. 12 · Triangular shape: multiply the diagonal, 4 times 3 is 12.
  5. Its rows are dependent and the system has no unique solution · Zero determinant means collapse: no inverse and no unique solution.
  6. 2 · Upper triangular, so multiply the diagonal: 1 times 1 times 2 is 2.
  7. 12 · Stack them as columns and take the absolute determinant: 3 times 4 minus 0 is 12.
  8. 12 · Triangular determinants multiply the diagonal: 4 times 3 = 12.
  9. Area 15 with orientation flipped · Size 3 triples areas, so 5 becomes 15, and the negative sign flips orientation.
  10. 2 · Upper triangular, so multiply the diagonal: 1 times 1 times 2 = 2.
Worksheet · LightMySky