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. Vectors (5, 2) and (4, 1) span a parallelogram. What is its area?

    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 but orientation flips

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

    • 7
    • 12
    • 21

  5. The rows of the matrix with rows (2, 4) and (3, 1) are swapped, and the swapped matrix has determinant 10. What was the determinant before the swap?

    • 10
    • -20
    • -10
    • 0

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

    Answer: ______________

  7. Vectors (1, 3) and (4, 2) span a parallelogram. What is its area?

    Answer: ______________

  8. The system x + 2y = 3 and 2x + 4y = 7 has coefficient determinant 0. What is true about its solutions?

    • Exactly one solution, because the two constants differ
    • No unique solution, and here the lines never meet
    • Exactly one solution, at the point (3, 0)
    • Infinitely many solutions, since the rows are multiples

  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 5 with orientation kept the same
    • Area 2 with orientation flipped

  10. Use cofactor expansion along row 1 to find the determinant of the matrix with rows (1, 2, 0), (3, 1, 2) and (0, 1, 1).

    Answer: ______________

LightMySky · lightmysky.comW1-mt_fWr2D_1BdM-s1

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. 3 · As columns they give rows (5, 4) and (2, 1). The determinant is 5 times 1 minus 4 times 2, which is minus 3, so the area is 3.
  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. -10 · A row swap flips the sign, so the original determinant is -10. Directly, 2 times 1 minus 4 times 3 is -10.
  6. 2 · Upper triangular, so multiply the diagonal: 1 times 1 times 2 is 2.
  7. 10 · Stack them as columns: rows (1, 4) and (3, 2). The determinant is 1 times 2 minus 4 times 3, which is minus 10, so the area is 10.
  8. No unique solution, and here the lines never meet · Determinant 0 rules out exactly one solution. Doubling the first equation gives 2x + 4y = 6, which clashes with 2x + 4y = 7, so the lines are parallel and there is no solution.
  9. Area 15 with orientation flipped · Size 3 triples areas, so 5 becomes 15, and the negative sign flips orientation.
  10. -7 · Along row 1: 1 times (1 times 1 minus 2 times 1), minus 2 times (3 times 1 minus 2 times 0), plus 0. That is -1 - 6 + 0 = -7.
Worksheet · LightMySky