**Igor Kriz**

**1.**

Find the inverse of the following matrix:

**2.**

Consider the following general problem: let ,..., be constants. Suppose a sequence is defined as follows: ,..., are given, and

This is called a *linear recursion*.

(a) Prove that if the polynomial

has *k* distinct roots (i.e. for
different), then any sequence of the form

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

for some *t* between *x* and *a*.

(a) Prove that for any number *x*,

(Recall that .)

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

for all .

**4.**

If a square matrix *A* satisfies for some *n*, prove that *I*-*A*
is invertible, where *I* is the identity matrix of the same dimension.
[Recall the formula for , and prove it for matrices.]

Wed Jan 28 21:55:04 EST 1998