96 CHAPTER 5. SOME IMPORTANT LINEAR ALGEBRA

Lemma 5.9.7 Suppose det(A) = 0. Then for all sufficiently small nonzero ε, det(A+ εI) ̸=0.

Proof: First suppose A is a p× p matrix. Suppose also that det(A) = 0. Thus, theconstant term of det(λ I−A) is 0. Consider εI+A≡ Aε for small real ε . The characteristicpolynomial of Aε is

det(λ I−Aε) = det((λ − ε) I−A)

This is of the form

(λ − ε)p +ap−1 (λ − ε)p−1 + · · ·+(λ − ε)m am

where the a j are the coefficients in the characteristic equation for A and m is the largest suchthat am ̸= 0. The constant term of this characteristic polynomial for Aε must be nonzero forall positive ε small enough because it is of the form

(−1)mε

mam +(higher order terms in ε)

which shows that εI +A is invertible for all ε small enough but nonzero.