Let A be a 2×2 matrix with exactly one real eigenvalue λ, and assume λ has geometric multiplicity 1 (note that this means A is not diagonalizable).
Prove that A−λI≠(0 0)
(0 0),
but (A−λI)² =(0 0) (Hint: Use problem 4.4.25 from Homework 8.)
(0 0)