Overview

Every invertibility test built across Topic 4 — trivial kernel, full rank, $n$ pivots — is a structural condition, checked by running an algorithm (elimination) to completion. This short topic supplies the one test that is instead a single number: the determinant $\det A$, which is nonzero exactly when $A$ is invertible, and whose magnitude has a clean geometric meaning as a volume scaling factor. The topic exists mainly to complete the invertibility chain with its last unproved link, and to build the one piece of machinery Stage 3 needs immediately: the characteristic polynomial $\det(A-\lambda I)$ that defines an eigenvalue.