Solving by area ratios, Cramer's rule
What you’ll learn
Section titled “What you’ll learn”This is lesson 12 of Track 4 (Visual Math: Linear Algebra). The inverses lesson said the solution to M · v = b is M^-1 · b and then deliberately did not compute it; this lesson computes it, with Cramer’s rule. By the end you will be able to solve a 2x2 system as a ratio of determinants, and, more importantly, explain exactly why that ratio is the answer. The whole rule falls out of one fact you already own from the determinant lesson: a linear transformation scales every area by its determinant. The point is not that Cramer’s rule is fast (it is not); it is that a method you might have met as a memorized recipe turns out to be a clean statement about how transformations stretch area. The source curriculum is 3Blue1Brown’s Essence of Linear Algebra by Grant Sanderson, freely available at 3blue1brown.com.
Where this fits
Section titled “Where this fits”This is lesson 12 of 15, the third lesson of Phase 3 (advanced perspectives). It closes a question the inverses lesson opened (how do you actually find the input that lands on b) using the area-scaling fact from the determinant lesson. It follows Deriving the 3D cross product in sequence, though it leans conceptually on the determinant and inverses lessons rather than on the cross product. The next lesson, Change of basis, makes the coordinate system itself the subject: the same vector has different coordinates in different bases.
Before you start
Section titled “Before you start”Prerequisites: the genuinely load-bearing lessons here are The determinant (a transformation scales area by det(M), the fact the whole derivation rests on) and Inverses, column space, and null space (the solve-the-system framing and the det(M) ≠ 0 condition). You should be comfortable computing a 2x2 determinant and reading it as a signed area. The practice is pen and paper.
About the math
Section titled “About the math”The arithmetic is a handful of 2x2 determinants per system. You will solve two systems by the column-replacement formula and check the answers, confirm the coordinate-is-an-area claim with a determinant, and see a zero-determinant system refuse to be solved. The computation is light; the depth is in the derivation, watching a coordinate equal an area and that area scale by det(M).
By the end, you’ll be able to
Section titled “By the end, you’ll be able to”- Solve a 2x2 linear system by the Cramer’s-rule column-replacement formula
- Explain why a coordinate of the unknown equals a signed area (the parallelogram with a basis vector)
- Derive each coordinate as det(M with its column replaced by b) divided by det(M), using area-scaling
- Explain why det(M) = 0 makes the rule fail and why that correctly signals no unique solution
- Recognize that Cramer’s rule is a conceptual and small-system tool, not the fast method for large systems
Time and difficulty
Section titled “Time and difficulty”- Read time: about 11 minutes
- Practice time: about 15 minutes (a solve-the-system exercise, a geometric-why and collapsed-case drill, and flashcards)
- Difficulty: standard (a Phase 3 lesson; light determinant arithmetic, with the depth in the area-ratio derivation)