MA-305 Homework 2

Due at 2:35 pm, Thursday, February 5, 1998

Calculations necessary for these problems may be done either by hand or with Maple. Solutions may be submitted in person in class, or you may email an ASCII text, Maple Worksheet (.mws), or html-formatted document to the TA, John Haws (jchaws@eos.ncsu.edu).

1. A homogeneous system of linear equations is a system in which the last column of the augmented matrix contains all zero entries, i.e. the right-hand side of each equation is zero. Explain why such a system always has a solution, and describe the geometric properties of the solution set.

2. Let
   
   

   A =	[ 2 -3 ] [ 4 1 ]

   Find
   1. 2A - 3A²
   2. A⁴ - 2A² + A - 4I₂
      

3. Let
   
   

   A =	[ 2 4 6 ] [ 1 1 1 ]	and	B =	[ 3 -3 1 ] [ -2 1 -1 ] [ 0 0 1 ]

   If possible, determine the following. If the calculation is not possible, explain why not.
   
   
   1. (AB)^T
   2. A^TB^T
   3. BB^T
   4. AA^T
   5. A^TA
   6. B^TA^T
      

4. Consider Fibonacci's famous problem under different conditions. Suppose you start with one pair of immature (newborn) rabbits in a closed pen. If it takes three months for the rabbits to mature, and one additional month to produce one pair of offspring, how many rabbits are present in the pen after 15 months?
   (Hint: Use matrix multiplication to relate f_i, the number of rabbits present after i months.)