MA-305 Homework 2

Solution

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.

   Solution:

   (0, 0, 0) is always a solution to a 3 unknown homogeneous linear system, so the solution set must pass through the origin all the time.

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

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

   Solution:

   (a)	[-3	-2]
   	[ 4	1]

   (b)	[-24	-30]
   	[ 60	36]

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.
   

   Solution:

   (Will be posted with the solution to Homework 3.)

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?
   

   Solution:

   (a) 200*220 + 150*300 + 120*120 = 103400
   (b) 50*100 + 40*120 + 25*250 = 16050

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?

   Solution:

   x=-1, y=2.

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