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courses › Linear Algebra

Inverses & rank

level 48 course

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

Pen and paper is fine · no calculator needed why?

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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

The inverse A⁻¹ undoes A, the way dividing undoes multiplying. A 2 × 2 matrix has one exactly when its determinant is not 0.

Cramer's rule solves two equations in x and y with three 2 × 2 determinants. The rank of a matrix is the most rows you can keep with none built from the others by multiplying and adding.

Techniques

2 × 2 inverse: swap, flip, divide

Any entry of the inverse of [[a, b], [c, d]].

  1. Find the determinant, ad − bc.
  2. Swap a and d, and change the signs of b and c: [[d, −b], [−c, a]].
  3. Divide the entry you need by the determinant, and reduce the fraction.
worked example

Example: A = [[2, 3], [4, 8]] (rows listed). What is the top-right entry of A⁻¹? Enter a fraction or a whole number.

  1. det = 2 × 8 − 3 × 4 = 16 − 12 = 4.
  2. Swap and flip: [[8, −3], [−4, 2]].
  3. Top-right: −3 ÷ 4 = −3/4.

Answer: −3/4

Cramer’s rule for two equations

Solving ax + by = e and cx + dy = f.

  1. D = ad − bc, from the numbers in front of x and y.
  2. x = (ed − bf) ÷ D, with the right-hand sides in the x-column.
  3. y = (af − ec) ÷ D, with them in the y-column.
  4. Check both values in both equations. Give x first.
worked example

Example: Solve 3x + 2y = 7 and x − y = −1. What is x?

  1. D = 3 × (−1) − 2 × 1 = −5.
  2. x = (7 × (−1) − 2 × (−1)) ÷ (−5) = −5 ÷ (−5) = 1.
  3. y = (3 × (−1) − 7 × 1) ÷ (−5) = 2. Both equations check.

Answer: 1

Rank: count the independent rows

The rank of a 3 × 3 or 3 × 4 matrix.

  1. Set aside any row that is a multiple of another, or a sum of multiples of the others.
  2. Repeat until no row left is built from the others. The rows left give the rank.
  3. None set aside: the rank is 3. For a 3 × 3, a nonzero determinant confirms it.
worked example

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

  1. Row 3 is twice row 1 plus row 2, so it adds nothing new.
  2. Rows 1 and 2 are not multiples of each other.
  3. So the rank is 2.

Answer: 2

Tips by skill

  • TipInverse entry: Swap a and d, change the signs of b and c, then divide the entry you need by ad − bc.
  • TipSolve a system: Use elimination or Cramer’s rule with D = ad − bc. Check both equations, and enter x first.
  • TipRank: Set aside any row that is a multiple, or a sum of multiples, of the others. Count the rows left.

Watch out for

  • Skipping a step of the inverse: swap a and d, change the signs of b and c, and divide by the determinant, sign included.
  • Giving y before x. Enter x first.
  • Answering 3 for the rank without checking whether a row is built from the others.

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.

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