MA-305 Homework 2

Due at 4:05pm, Thursday, September 11, 1997

You may do the calculations necessary for these problems either by hand or with Maple. Please submit a handwritten solution, or email an ASCII/Postscript/html document to the TA. (You may also submit through email which attached your Maple worksheet.)

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 sides of the equations are all zero. Consider a system in 3 unknowns, the coordinate values of 3d-space. Explain why the solution set always must pass through the origin.

2. *Let	A=	[ 1	-1]	. Find
   		[ 2	3]

   (a) A²-2A.
   (b) 3A³-2A²+5A-4I₂.
   

3. [2

   -1]

   *Let

   A=

   [ 2

   1

   -2]

   ,

   B=

   [3

   4]

   [ 3

   2

   5]

   [1

   -2]

   If possible, compute:

   (a) (AB)^T.
   (b) B^TA^T.
   (c) A^TB^T.
   (d) BB^T.
   (e) B^TB.
   

4. *A photography business has a store in each of the following cities: New York, Denver, and Los Angeles. A particular make of camera is available in automatic, semiautomatic, and nonautomatic models. Moreover, each camera has a matched flash unit and a camera is usaully sold together with the corresponding flash unit. The selling prices of the cameras and flash units are given (in dollars) by the matrix

   		Automatic	Semiautomatic	Nonautomatic
   A=	[	200	150	120]	Camera
   	[	50	40	25]	Flash unit

   The number of sets (camera and flash unit) available at each store is given by the matrix

   		New York	Denver	Los Angeles
   	[	220	180	100]	Automatic
   B=	[	300	250	120]	Semiautomatic
   	[	120	320	250]	Nonautomatic

   (a) What is the total value of the cameras in New York?
   (b) What is the total value of the flash units in Los Angeles?
   

5. Suppose the sequence of Fibonacci numbers have been started with values different than 0 and 1, namely with f₀ = x and f₁ = y. With these new initial values, one obtains f₈ = 29 and f₉ = 47. What are x and y?

(Starred problems are from ``Introductory Linear Algebra with Applications'' by Kolman and Hill)