References: Cramer's rule

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Linear Algebra, Chapter 12: "Cramer's rule, explained geometrically"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/cramers-rule
  Series index: https://www.3blue1brown.com/?topic=linear-algebra
  License: copyright Grant Sanderson; videos published on his site and YouTube
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Watch this next

• Cramer’s rule, explained geometrically (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. The key step, seeing the unknown coordinate as a signed area and watching that area scale by the determinant under the transformation, is much more convincing in motion than on the page. About thirteen minutes, and it makes the column-replacement formula feel inevitable rather than arbitrary.

Going deeper

• Essence of Linear Algebra (full series) by 3Blue1Brown. The series this track follows. The determinant chapter supplies the area-scaling fact this whole derivation rests on, and the inverses chapter posed the solve-the-system question this lesson answers; the next (Change of basis) asks what happens to a vector’s coordinates when you change the basis you measure it against.

• Khan Academy: Linear algebra for a slower, exercise-driven treatment of solving linear systems, including Cramer’s rule and the faster elimination methods, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track.

• Inverses, column space, and null space (earlier lesson). That lesson said the solution to M · v = b is M^-1 · b and that it exists only when det(M) != 0. Cramer’s rule is one concrete way to compute that solution, and its division by det(M) is the same invertibility condition seen again.

• Change of basis (next lesson). This lesson treated [x, y] as coordinates in the standard basis without comment. The next lesson makes the basis itself the subject: the same vector has different coordinates in different bases, and a change-of-basis matrix translates between them.