a. Let J be a k k Jordan block with eigenvalue . Show that (J I)k = O. b. (Cayley-Hamilton Theorem) Let A be an n n matrix, and let p(t) be its characteristic polynomial. Show that p(A) = O. (Hint: Use Theorem 1.5.) c. Give the polynomial q(t) of smallest possible degree so that q(A) = O. This is called the minimal polynomial of A. Show that p(t) is divisible by q(t).

L11 - 3 Inﬁnite Limits at Inﬁnity Def. x→∞m f(x)= ∞ if we can make f(x)a sleaswe want by choosingx suﬃciently large. We have similar deﬁnitions for lf i(x)= ∞, x→−∞ lim f(x)= −∞ and lim f(x)= −∞. x→∞ x→−∞ To evaluate limits at...