courses › Linear Algebra

Invert a matrix, solve with it, and measure its rank.

Pen and paper is fine · no calculator needed why?

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Opens at level 48.

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Builds on: Matrices (not open yet)

skills · practice stats

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

• Inverse entry par 30 s not tried yet

  worked example

  A = [[7, 7], [−5, 2]] (rows listed). What is the bottom-left entry of A⁻¹? Enter a fraction or a whole number.

  Answer: 5/49

  1. det = ad − bc = 7 × 2 − 7 × (−5) = 49.
  2. A⁻¹ = (1/49)[[2, −7], [5, 7]]: swap a and d, change the signs of b and c.
  3. So the bottom-left entry is 5/49.

• Solve a system par 45 s not tried yet

  worked example

  Solve −4x − y = −19 and 2x − 4y = 50. Enter x, y.

  Answer: (7, -9)

  1. Cramer’s rule. det = (−4) × (−4) − (−1) × 2 = 18.
  2. x = ((−19) × (−4) − (−1) × 50) ÷ 18 = 126 ÷ 18 = 7.
  3. y = ((−4) × 50 − (−19) × 2) ÷ 18 = −162 ÷ 18 = −9.

• Rank par 40 s not tried yet

  worked example

  What is the rank of [[0, 0, 2], [−1, 5, 6], [2, 4, 4]]?

  Answer: 3

  1. det = −28, not 0, so no row is a combination of the others.
  2. The rank is 3.

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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