7.2-02 Characteristic polynomials

Characteristic polynomials

Practise characteristic polynomials with the explicit monic convention det(tI−A). Work through diagonal, triangular, coupled two-by-two and block-structured examples while checking signs and coefficients.

What you will practise

• Construct the shifted matrix
• Expand a small determinant
• Use triangular and block structure
• Test a prescribed eigenvalue in a parameter family

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review small determinants, polynomial expansion and factoring. 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 more demanding items combine block structure with a nontrivial two-by-two calculation. 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.