16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 6.2.8

This problem hinges on the Sylvester determinant identity and the deep symmetry between $AB$ and $BA$. The key insight is that invertibility of $I - AB$ and $I - BA$ are actually equivalent—they fail together or succeed together—even though the matrices have different sizes in the block sense. Rather than computing inverses directly, construct the claimed inverse by multiplying it against $I - BA$ and verifying you get the identity; the algebra works because $AB$ and $BA$ share the same nonzero eigenvalues, and the formula respects this hidden symmetry.

This problem hinges on the Sylvester determinant identity and the deep symmetry between $AB$ and $BA$. The key insight is that invertibility of $I - AB$ and $I - BA$ are actually equivalent—they fail together or succeed together—even though the matrices have different sizes in the block sense. Rather than computing inverses directly, construct the claimed inverse by multiplying it against $I - BA$ and verifying you get the identity; the algebra works because $AB$ and $BA$ share the same nonzero eigenvalues, and the formula respects this hidden symmetry.