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Inverses & rank

lesson · about 3 minutes

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

Pen and paper is fine · no calculator needed why?

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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

watch out for

practice

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Inverse entry

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

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

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.

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