4.3-02 Inverses, powers and scalar factors

Inverses, Powers and Scalar Factors

Combine determinant identities accurately when expressions contain inverses, powers and scalar multiples. Begin with reciprocals and powers, then practice expressions where the matrix dimension controls the scalar contribution. Later items require solving for an unknown determinant or scalar.

What you will practise

• Evaluate determinant expressions involving inverse powers.
• Track the dimension when an entire matrix is scaled.
• Handle odd and even powers without losing sign possibilities.
• Solve determinant equations with stated real-number assumptions.

Prerequisites and study approach

Useful preparation: Product determinants, inverses, exponents and elementary algebra. 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 scaling a whole matrix as if only one row changed, or discarding a possible sign after taking an even power. 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.