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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 6 "Matrix" }{TEXT 256 37 " Algebra \+
and the Algebra of Numbers I" }}{PARA 257 "" 0 "" {TEXT 260 21 "- A co
mparative study" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT 258 23 "Linear Algebra Project " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "
" }{TEXT 259 10 "Objective:" }}{PARA 0 "" 0 "" {TEXT -1 154 "We find o
ut what properties of matrix addition and multiplication are the same \+
as those of numbers, and which ones are different by doing an experime
nt.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 280 
"As you get into doing this experiment, you will see that it would be \+
almost impossible to do it without using a computer. Spending a lot of
 time doing simply arithmetic in matrix multiplication will divert you
r attention from discovering basic laws of matrix algebra for yourself
." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 0 "" }{TEXT 261 16 "Getting started:" }}{PARA 0 "" 0 "" {TEXT -1 
56 "We will need several matrices with which to experiment. " }}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 262 36 "Let us construct matric
es A, B and C" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(linalg
);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A:= matrix(4,4,[[1,
0,-1,3],[2,1,4,-2],[0,-5,0,1],[-1,2,-1,3]]);\n" }}}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "B:= matrix(4
,4,[[2,1,0,7],[-2,5,1,2],[4,1,3,-6],[-1,1,-8,2]]);\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 66 "C:= matrix(4,4,[[2,1,-5,-2],[-3,4,-1,3],[
0,2,0,1],[7,13,-52,3]]);\n" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "
" }{TEXT 263 12 "Experiments:" }{TEXT -1 0 "" }}{SECT 0 {PARA 20 "" 0 
"" {TEXT 268 0 "" }{TEXT 264 38 "How to multiply two matrices in Maple
:" }}{PARA 0 "" 0 "" {TEXT -1 112 "To multiply matrix A and B you can \+
either use the \"multiply(A,B)\" command or the \"evalm(A&*B)\". Let u
s try both." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "AB:=multiply(A,B);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "AB:= evalm(A&*B);\n" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 267 9 "Problems:" }}
{PARA 0 "" 0 "" {TEXT -1 85 "1. Compute AB and BA. Compare your answer
s. What can you deduce from this comparison?" }}{PARA 0 "" 0 "" {TEXT 
-1 102 "    Without using a computer try to find two 2x2 matrices X an
d Y where neither are identity matrices " }}{PARA 0 "" 0 "" {TEXT -1 
17 "     and XY = YX." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 110 "2. Compute A(BC) and (AB)C. Compare your answers. Wha
t property of matrix multiplication does this illustrate?" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "3. The three matri
ces that you worked with were given to you by me.  May be I rigged the
m to come " }}{PARA 0 "" 0 "" {TEXT -1 133 "    out in a certain way! \+
 To test for yourself take three random matrices R, S and T     [Check
 below to construct random matrices.]" }}{PARA 0 "" 0 "" {TEXT -1 90 "
(a) Try to see if you can come up with three matrices where R(ST) and \+
(RS)T are not equal." }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 
265 33 "How to construct random matrices:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 266 68 "Maple choses entries of its random matrices from t
he set (-99, 99). " }}{PARA 0 "" 0 "" {TEXT -1 94 "Just give the dimen
sions of the matrix and Maple will give you a random matrix. Here they
 are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "R:= randmatrix(3,4);\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 77 "Now if you execute the command again you will get a diffe
rent matrix. Try it!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "S:=
randmatrix(4,2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "T:= r
andmatrix(2,5);\n" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 
5 "" 0 "" {TEXT -1 1 " " }{TEXT 269 8 "Problem:" }}{PARA 0 "" 0 "" 
{TEXT -1 203 "4.  Compute A(B+C) and AB + BC.  Compare your answers.  \+
What property have you illustrated? [Matrices A and B were given at th
e start of the experiment. You do not have to retype them-- Maple reme
mbers.]" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 
74 "__________________________________________________________________
________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
93 "MSIP Grant #120A80089-98: \"Three Urban Calculus Reform Programs: \+
Adopting the Best\" 1998-2001" }}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }