7.3-03 Powers, inverses and shifts

Powers, inverses and shifts

Follow a known eigendirection through powers, inverses, scalar shifts and polynomial expressions. Learn when scalar shortcuts are valid and why a transformed matrix can have additional eigendirections.

What you will practise

• Apply polynomial expressions to an eigenvector
• Check invertibility before taking reciprocals
• Handle common eigenvectors of two matrices
• Diagnose failures of reverse implications

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix powers, inverses and the eigenvector equation. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently.

The final items combine several operations and identify parameter shifts that make a matrix singular. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.