doing math

courses › Linear Algebra

Matrices

level 42 course

Multiply matrices and compute determinants.

Pen and paper is fine · no calculator needed why?

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Opens at level 42.

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Builds on: Vectors (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

A matrix is a grid of numbers. Here it is written as a list of rows: [[1, 2], [3, 4]] has first row (1, 2) and second row (3, 4).

Each entry of a product AB is a dot product: a row of A with a column of B. So AB and BA are usually different.

The determinant of a square matrix measures how much it stretches area or volume; a minus sign means it also flips them over. It is 0 exactly when the matrix squashes space flat.

Techniques

Row times column

One entry of a matrix product AB.

  1. For the entry in row i, column j, take row i of A and column j of B.
  2. A column runs down: column j holds the j-th entry of each row.
  3. Multiply matching entries and add, as in a dot product.
worked example

Example: A = [[2, −1], [0, 3]] and B = [[4, 1], [5, −2]] (rows listed). What is the entry in row 1, column 2 of AB?

  1. Row 1 of A is (2, −1). Column 2 of B is (1, −2).
  2. 2 × 1 + (−1) × (−2) = 2 + 2 = 4.

Answer: 4

2 × 2 determinant: ad − bc

Any 2 × 2 matrix [[a, b], [c, d]].

  1. Multiply down the main diagonal, a × d.
  2. Multiply the other diagonal, b × c.
  3. Subtract the second from the first. Keep the signs of negative entries.
worked example

Example: What is the determinant of [[3, −2], [5, 4]]?

  1. ad = 3 × 4 = 12.
  2. bc = (−2) × 5 = −10.
  3. 12 − (−10) = 22.

Answer: 22

3 × 3: expand along the zeros

A 3 × 3 determinant, especially one with zeros in it.

  1. Pick the row or column with the most zeros.
  2. Signs follow a checkerboard: + − + on the top row, − + − in the middle, + − + at the bottom.
  3. Each nonzero entry gives sign × entry × the 2 × 2 determinant left when you cross out its row and column.
  4. Add those terms. The zero entries drop out.
worked example

Example: What is the determinant of [[0, 3, 0], [2, 1, 4], [1, 5, 3]]?

  1. Row 1 has two zeros. Its middle position has sign −.
  2. Cross out row 1 and column 2: [[2, 4], [1, 3]].
  3. That determinant is 6 − 4 = 2.
  4. det = −3 × 2 = −6.

Answer: −6

Tips by skill

  • TipMatrix product entry: Take row i of A and column j of B. Multiply matching entries and add.
  • Tip2 × 2 determinant: Main diagonal product minus the other diagonal product: ad − bc. Keep the signs of negative entries.
  • Tip3 × 3 determinant: Expand along the row or column with the most zeros, using the + − + checkerboard signs and each entry’s 2 × 2 determinant.

Watch out for

  • Multiplying the entries in the same spot of A and B. An entry of AB is a whole row times a whole column.
  • Adding the two diagonal products. The determinant subtracts them.
  • Forgetting the checkerboard sign. The middle entry of the top row carries a minus.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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