MATH 296 PROBLEMS 4

IGOR KRIZ

1. Find the inverse of the following matrix:

( \begin{matrix}−1 & 4 & 5 & 2 \\ 0 & 0 & 0 & −1 \\ 1 & −2 & −2 & 0 \\ 0 & −1 & −1 & 0 \end{matrix} )

2. Consider the following general problem: let a_{0},...,a_{k−1} be constants. Suppose a sequence (x_{n}) is defined as follows: x_{0},...,x_{k−1} are given, and

x_{n} = a_{k−1}x_{n−1} + ... + a_{0}x_{n−k} for n ≥ k. (1)

This is called a linear recursion.

(a) Prove that if the polynomial

p(x) = x^{k} − a_{k−1}x^{k−1} − a_{k−2}x^{k−2} − ... − a_{0}x^{0}

has k distinct roots (i.e. p(x) = (x − γ_{1})...(x − γ_{k}) for γ_{1}, ..., γ_{k} different), then any sequence of the form

x_{n} = A_{1}γ_{1}^{n} + A_{2}γ_{2}^{n} + ... + A_{k}γ_{k}^{n},

A_{i} constant, satisfies (1).

(b) Using the method from (a), solve the following problem: In a biological experiment, a culture of cells is grown in a test tube. Every cell present on day n divides into 4 cells on day n + 1, but as a byproduct of its growth produces an amount of toxin which kills exactly one cell on day n + 2. If there was only one cell on day 1, how many cells are there on day n? (4 on day 2, 4.4−1 = 15 on day 3, 15.4−4 = 54 on day 4, e.t.c.).

3. Recall the Taylor expansion formula from Problem set 2: If f is a real function which is defined and has n + 1 derivatives in an interval containing a, x, then

f(x) = f(a) + \frac{f′(a)}{1!}(x − a) + ... + \frac{f^{(n)}(a)}{n!}(x − a)^{n} + \frac{f^{(n+1)}(t)}{(n + 1)!}(x − a)^{n+1}

for some t between x and a.

(a) Prove that for any number x,

lim_{n→∞} \frac{x^{n}}{n!} = 0.

(Recall that n! = 1.2.3. . . . .n.)

(b) Using (a), and the Taylor expansion formula at a = 0, prove that

e^{x} = lim_{n→∞}(1 + \frac{x}{1!} + ⋯ + \frac{x^{n}}{n!} )

for all x ∈ ℝ.

4. If a square matrix A satisfies A^{n} = 0 for some n, prove that I −A is invertible, where I is the identity matrix of the same dimension. [Recall the formula for \frac{1−x^{n}}{1−x}, and prove it for matrices.]