Homework #2
> 
 Name:

Prefered Email:

Collaborators: 

Due February 19th  at  23:45:00.  Submit electronically via Wolfware @ http://submit.ncsu.edu/.   Neither email nor paper submissions will be accepted.
> 
> #Hint: Make liberal use of evalm() 
> 
> with(linalg):
Warning, the protected names norm and trace have been redefined and unprotected

Problem the first:

a) For which value(s) of 'a' does the following matrix have rank 2? Explain.
                 [ 1       2        3      4]
                 [ 0    (a+1)   a     1]
                 [ 0        a        a     0]
> 
> 
> 
> 
b) For which values of 'a' is the following  matrix in row echelon form? Explain.
                 [ 1          2           3          4]
                 [ 0      (a-1)       a         1]
                 [ 0      a^2-1     a-1    0]
> 
Given the matrix
         	 [  1/2   6    1  ]
  A =	 [   2    0    3   ]
	          [   4    1    0   ]

c) Compute A', the transpose of A.
> 
> 
> 
> 
> 
 Problem the second:

a) Is it true that for all square matrices A, B that (AB)^2 = A^2 B^2?  Explain your answer, and give examples/counter-examples.

>   
> 
> 
> 
b) Given the binary operation [,]:R^(n,n) x R^(n,n) -> R^(n,n) defined by [A,B] = A*B - B*A decide whether this operation is commutative. (i.e. if it is, show it algebraically, otherwise give an example which shows it is not).
> 
> 
> 
[Bonus]  Is the operation in part (b) associative? (if it is, show it algebraically, otherwise give an example which shows it is not).
>  
> 
> 
> 
> 
> 
 Problem the third:

Suppose the after birth, the Fibonacci rabbits take 2 months to be able breed (instead of just one), and that each pair gives birth to 2 new pairs of rabbits each month there after.

a) If we start with 1 pair of baby rabbits at month 0, how many rabbits in months 1-5?
> 
> 
> 
> 
> 
b) Write a formula for the number of rabbits in month n in terms of the number of rabbits in previous months (n >= 3).
> 
> 
> 
> 
c) Write a matrix (it should be 3x3) which will compute the population at month n given the population in the previous three months.Use matrix powering to compute the population in month 20.
> 
> 
Problem the fourth:

Given a polynomial f(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_2 x^2 + a_1 x + a_0 (where a_i's are real numbers) we define the polynomial evaluated on the m x m square matrix B as follows: f(B) = a_n * B^n + ... + a_2 * B^2 + a_1 * B + a_0* I_m.
 Here I_m is the m by m identity matrix.

e.g. If f(x) = 3*x^2 - 6, then f(A) = 3*A^2 - 6* I_m

If  B = matrix([[0, 0, -4], [1, 0, 1], [0, 1, 0]]):

a) Compute g(B), where g(x) = x^2 - x + 1

> 
> I_3 := band([1],3);
> 
> 
b) Compute h(B), where h(x) = 3*x^3 - 3*x + 12
> 
> 
> 
> 
 If C = [ 0  -4 ]   
             [ 1     2  ]

c) Find a polynomial f(x) =  x^2 + b x + c with f(C) = 0^(2x2) (Hint: write down f(C) as a matrix with b and c constants, Maple will do this for you quite easily). Can you think of another polynomial which when evaluated on C would give 0_(2x2)? 
> 
> I_2 := band([1],2);
> 
> 
> 
 Problem the last:

Given the matrix
         	 [  1/2   6    1  ]
  A =	 [   2    0    3   ]
	          [   4    1    0   ]

a)  Is it true that B^(n+m) = B^n B^m for any square matrix B, and all integers n and m?  If it is not, provide a counter example.  If it is, sketch a short argument why. 
> 
> 
> 
> 
b) Compute A^2, A^4, A^6, and A^7 (in that order) using exactly one matrix multiplication for each (this means no using ^ either, though you may reuse matrices you've already computed);
> 
> 
> 
> 
> 
d) [More Difficult, don't worry too much if you can't get it.] In class we talked about how to compute A^(2^k) efficiently.  Consider the binary representation of an integer: 

      m = b0 + b1 * 2 + b2 * 2^2 + ... + bn * 2^n  where each bi is 0 or 1. (i.e. (bn b(n-1) .... b1 b0) are the binary digits of m)
 
Keeping in mind parts (b) and (c), how might you adapt the method from class to compute A^m?
> 
> 
> 
> 
> 
[Bonus] In the case where m = 31, compare the number of operations it takes to compute A^m your way versus computing A*A*A*...*A.
> 
> 
> 
> 
[Bonus] Implement this method in Maple. The following is a Maple function to get a list of the binary digits of m (not guaranteed to be good code, but might be useful):
> 
> binaryDigits := proc(m)
> # Call as: binaryDigits(yournumber);
> local binary_rep, binary_digits, bit;
> binary_rep := convert(m,binary); # Maple converts the number to binary
>                                  # Note that it still treats binary_rep as
>                                  # as a decimal number!
> 
> binary_digits := [];             # Initialize
> while binary_rep >= 1 do
>   bit := irem(binary_rep,10);    # Grab the last 1 or 0
>   binary_digits := [op(binary_digits),bit];  # Store the bit in our list
>   binary_rep := (binary_rep - bit) / 10;      # Strip the last bit, and shift
> od;
> RETURN(binary_digits); # Note that the digits are returned in reverse order
> end:
> 
> 
>