3Blue1Brown: Matrix Multiplication as Composition - Chapter 4 Quiz

Matrix multiplication as composition

This quiz accompanies Chapter 4 of 3Blue1Brown’s Essence of Linear Algebra. It practices reading products as sequences of linear transformations and connecting algebraic rules with geometric effects.

Study focus

• Read products from right to left when they act on column vectors.
• Construct product columns by transforming the right-hand matrix’s columns.
• Distinguish associativity from commutativity.
• Interpret repeated projections, reflections, identity maps and inverses.
• Calculate simple composite maps and determinant effects.

For compatible matrices, $$ABv=A(Bv)$$ and $$A(BC)=(AB)C$$. The order of factors usually matters, although particular pairs may commute. For square matrices, $$\det(AB)=\det(A)\det(B)$$.

Familiarity with vector coordinates, linear combinations and matrix-vector multiplication is recommended. Some items require several selections. Trace basis vectors or calculate a small example before reading the explanation. Pay attention to stated invertibility and nonzero-vector assumptions. An inverse must exist before it can undo a map, and a fixed vector is an eigenvector only when it is nonzero. The quiz includes introductory applications beyond the chapter’s main composition theme.