jf A = [aii] is an II x II matrix, then the Irace of A. Tr(A). is defined as the sum of all elements on the main . diagolwl of A. Tr(A) = LUil Show each of the follow- ;=1 mg: (a) Tr(cA) = c Tr(A). where c is a real number (b) Tr(A + B ) = Tr(A) + Tr( B ) (e) Tr(AB) = Tr(BA) (d) Tr(AT) = Tr(A) (e) Tr(AT A) 2:

Lecture 27 L’Hˆopital’s Rule (Section 4.5) Consider the following limits: e − 1 ex. lim x→0 x ln(x) ex. x→∞mx +1 These limits are Indeterminate Forms and ∞ .e 0 ∞ have seen how to evaluate some limits of this form, but the process can be complicated. √ 2 ex. lim x +3 − 2 x→1 x − 1 sinx lim x→0 x