3.5-06 Inverses of products

Inverses of products — linear algebra practice

Reverse the order of operations to derive inverses of composed transformations. Verify two- and three-factor rules, calculate a product inverse and solve an equation with unknown matrix between two factors. Contrast valid rules with the false idea that inverses distribute across addition.

What you will practise

• Undo composed maps in reverse order
• Verify inverse identities
• Solve left-and-right matrix equations

This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies.

Before starting, review inverses, associativity and noncommutativity. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to manipulate reversible matrix expressions.