4.3-01 Determinant of a product

Determinants of Matrix Products

Connect composition of linear transformations with determinant multiplication. Practice numerical identities, orientation and volume factors, recovery of an unknown factor and consequences for invertibility. Boundary examples distinguish square factors from rectangular matrices and ordinary products from entrywise multiplication.

What you will practise

• Compute determinant values for composed square maps.
• Use product information to reason about invertibility.
• Recognize where rectangular dimensions invalidate a familiar rule.
• Explain determinant invariance under an invertible coordinate change.

Prerequisites and study approach

Useful preparation: Matrix multiplication, invertibility and area or volume scaling. 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 assuming matrices commute because their determinant values do, or confusing entrywise multiplication with a matrix product. 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.