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Math 296 problems 4

Igor Kriz


Find the inverse of the following matrix:



Consider the following general problem: let tex2html_wrap_inline87 ,..., tex2html_wrap_inline89 be constants. Suppose a sequence tex2html_wrap_inline91 is defined as follows: tex2html_wrap_inline93 ,..., tex2html_wrap_inline95 are given, and


This is called a linear recursion.

(a) Prove that if the polynomial


has k distinct roots (i.e. tex2html_wrap_inline103 for tex2html_wrap_inline105 different), then any sequence of the form


tex2html_wrap_inline109 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 tex2html_wrap_inline147 .)

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


for all tex2html_wrap_inline153 .


If a square matrix A satisfies tex2html_wrap_inline157 for some n, prove that I-A is invertible, where I is the identity matrix of the same dimension. [Recall the formula for tex2html_wrap_inline165 , and prove it for matrices.]

Igor Kriz
Wed Jan 28 21:55:04 EST 1998