Matrix multiplication and what it represents · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Multiplying matrices in the right order

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

  1. A has rows (3, 1) and (2, 4), and B has rows (2, 5) and (6, 1). What is the top left entry of AB?

    Answer: ______________

  2. A has columns (1, 0) and (2, 1), and v has entries 3 and 4. What is the bottom entry of Av?

    Answer: ______________

  3. A has rows (1, 0) and (2, 1), and B has rows (3, 4) and (5, 6). What is the top right entry of AB?

    Answer: ______________

  4. A is 2 by 3 and B is 3 by 4. What shape is the product AB?

    • 2 by 4
    • 3 by 3
    • 3 by 4

  5. A is a 2 by 2 matrix and v holds 2 entries. How is the product Av best read?

    • A sum of scaled rows from A
    • The diagonal entries of A
    • A sum of scaled columns from A

  6. A has rows (2, 1) and (0, 3), and B has rows (4, 5) and (6, 7). What is the bottom left entry of AB?

    Answer: ______________

  7. Let A have rows (1, 2, 0) and (0, 1, 3), and let v be (2, 4, 5). What is Av?

    • (10, 19)
    • (6, 17)
    • (14, 12)
    • (7, 9)

  8. A is a 3 by 2 matrix. Which product with A on the left is defined?

    • A times a 3 by 5 matrix
    • A times a 2 by 5 matrix
    • A times a 5 by 2 matrix

  9. A has rows (1, 2) and (0, 1), and B has rows (2, 0) and (3, 1). What is the top left entry of BA?

    • 2
    • 7
    • 8

  10. A has columns (2, 0) and (0, 2), and v holds entries 3 then 4. What is the top entry of Av?

    Answer: ______________

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Answer key

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

Multiplying matrices in the right order W1-mt_0fJOMPlCxr-s1

  1. 12 · Pair row 1 of A with column 1 of B: 3 times 2 plus 1 times 6 is 12.
  2. 4 · The product is 3 times (1, 0) plus 4 times (2, 1), which is (11, 4). The bottom entry is 4.
  3. 4 · Use row 1 of A, (1, 0), and column 2 of B, (4, 6): 1 times 4 plus 0 times 6 equals 4.
  4. 2 by 4 · The inner sizes agree at 3, so the outer pair gives 2 by 4.
  5. A sum of scaled columns from A · Each entry of v scales one column of A, and the scaled columns add to Av.
  6. 18 · Use row 2 of A, (0, 3), and column 1 of B, (4, 6): 0 times 4 plus 3 times 6 equals 18.
  7. (10, 19) · Av is 2 times column 1 plus 4 times column 2 plus 5 times column 3: (2, 0) + (8, 4) + (0, 15) = (10, 19).
  8. A times a 2 by 5 matrix · A has 2 columns, so the matrix on the right must have 2 rows. Only the 2 by 5 matrix fits.
  9. 2 · For BA, use row 1 of B, (2, 0), with column 1 of A, (1, 0): 2 times 1 plus 0 times 0 equals 2.
  10. 6 · Weight the columns: 3 times (2, 0) plus 4 times (0, 2) is (6, 8).
Worksheet · LightMySky