4.3-04 Multilinear is not linear

Why Multilinear Does Not Mean Linear

Diagnose false determinant additivity claims and see where the missing terms come from. Work with explicit counterexamples, row-by-row expansions and parameter-dependent matrices. The questions connect abstract multilinearity to concrete algebra that can be checked directly.

What you will practise

• Disprove whole-matrix additivity with small examples.
• Expand changes in two columns and track mixed terms.
• Distinguish a single-entry change from whole-matrix variation.
• Explain degree reduction when two changing columns share one direction.

Prerequisites and study approach

Useful preparation: Determinant axioms, two-by-two formulas and polynomial expansion. 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 dropping mixed terms, assuming a singular sum from singular summands or extending a one-row rule to the whole matrix. 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.