4.4-02 Geometric proof of Cramer's rule

A Geometric Proof of Cramer’s Rule

Explain the determinant-ratio formula through column combinations and signed volume. These questions build the proof in small steps, identifying which multilinear terms survive and why the others vanish. Geometric examples then connect the remaining term to a solution coordinate.

What you will practise

• Express the right-hand side as a combination of columns.
• Use repeated-column cancellation to isolate a coefficient.
• Interpret a coordinate as a signed area or volume ratio.
• Separate a valid determinant identity from permission to divide.

Prerequisites and study approach

Useful preparation: Cramer’s rule, multilinearity, dependent columns and signed area or volume. Work through the questions from foundational interpretation to more demanding applications. Both single-select and multiselect items are included; in a multiselect item, judge every statement independently. Try a calculation, sketch or short justification before consulting the explanation.

The worked feedback develops the reasoning and addresses common errors such as using unsigned ratios, assuming orthogonality is required or dividing by a determinant that vanishes. Use it to compare your method with the stated assumptions, then revisit any step you could not justify. This microlearning belongs to Module 4, Determinants, Volume and Orientation, in the Complete Linear Algebra Course.