3.5-03 The two-by-two inverse

The two-by-two inverse — linear algebra practice

Derive and apply the small-matrix inverse formula with its nonzero-denominator condition. Practice integer, fractional and negative-denominator examples, parameter exclusions and a system solve. The derivation connects cancellation in the product to the geometric obstruction when the denominator vanishes.

What you will practise

• Use and justify the inverse formula
• Check singular cases and parameters
• Solve and verify two-variable systems

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 two-by-two multiplication and equation solving. 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 prepare for systematic inversion by elimination.