Find the inverse of the following matrix:
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.).
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 .
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.]