7.2-01 From eigenvectors to a singular system

From eigenvectors to a singular system

Derive eigenvalue calculations from the nonzero-vector requirement rather than memorizing a determinant rule. Shift a matrix by a scalar identity and connect the resulting homogeneous system to singularity and rank.

What you will practise

• Rearrange the eigenvector equation
• Form a shifted matrix correctly
• Apply the nontrivial-kernel criterion
• Relate nullity to eigenspace dimension

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix subtraction, determinants, rank and null spaces. 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 item compares characteristic-polynomial sign conventions without changing their roots. 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.