{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 
11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 
0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Tim
es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 258 1 
{CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 
0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 49 "Maple Linear Algebra wo
rksheet for row operations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 38 "First call the Linear Algebra package." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "On page 7 of Lay, three elementary
 row operations are given: " }}{PARA 0 "" 0 "" {TEXT -1 3 "1. " }
{TEXT 260 11 "Replacement" }{TEXT -1 69 "- Replace one row by the sum \+
of itself and a multiple of another row." }}{PARA 0 "" 0 "" {TEXT -1 
3 "2. " }{TEXT 261 11 "Interchange" }{TEXT -1 24 "- Interchange two ro
ws. " }}{PARA 0 "" 0 "" {TEXT -1 3 "3. " }{TEXT 262 7 "Scaling" }
{TEXT -1 54 "- Multiply all entries in a row by a nonzero constant." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "In this
 worksheet we will illustrate the three elementary row operations, the
n use them to put a matrix into " }{TEXT 257 12 "Echelon Form" }{TEXT 
-1 7 ", then " }{TEXT 258 20 "Reduced Echelon Form" }{TEXT -1 29 " (se
e page 14 of Lay) by the " }{TEXT 259 23 "Row Reduction Algorithm" }
{TEXT -1 46 " (see page 17 of Lay) and one additional step." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 263 9 "Example 1" }
{TEXT -1 37 "- the three elementary row operations" }}{PARA 0 "" 0 "" 
{TEXT -1 24 "First define a matrix A." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "A := <<1,2,3>|<4,5,6>|<7,8,9>>;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")K.l8-%'MATRIXG6#7%7%\"\"\"\"\"%\"
\"(7%\"\"#\"\"&\"\")7%\"\"$\"\"'\"\"*%'MatrixG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 66 "Replacement- for the matrix A, replace row 1 with (r
ow 1+(5)row2)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "RowOperat
ion(A, [1,2], 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")cV
17-%'MATRIXG6#7%7%\"#6\"#H\"#Z7%\"\"#\"\"&\"\")7%\"\"$\"\"'\"\"*%'Matr
ixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Interchange- for the matri
x A, switch rows 2 and 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "RowOperation(A, [2, 3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTA
BLEG6%\")kv!\\\"-%'MATRIXG6#7%7%\"\"\"\"\"%\"\"(7%\"\"$\"\"'\"\"*7%\"
\"#\"\"&\"\")%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Scaling
- for the matrix A, multiply row 2 by the constant 0.5." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "RowOperation(A, 2, 0.5);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")Wb#\\\"-%'MATRIXG6#7%7%\"\"\"
\"\"%\"\"(7%$\"#5!\"\"$\"#DF2$\"#SF27%\"\"$\"\"'\"\"*%'MatrixG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 264 10 
"Example 2a" }{TEXT -1 35 "- put a matrix into an echelon form" }}
{PARA 0 "" 0 "" {TEXT -1 42 "Next define matrix B (see page 17 of Lay)
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "B := <<0,3,3>|<3,-7,-9
>|<-6,8,12>|<6,-5,-9>|<4,8,6>|<-5,9,15>>;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\")w$\\F\"-%'MATRIXG6#7%7(\"\"!\"\"$!
\"'\"\"'\"\"%!\"&7(F/!\"(\"\")F3F6\"\"*7(F/!\"*\"#7F9F1\"#:%'MatrixG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "RowOperation(B, [1, 3]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")GVy7-%'MATRIXG6#7%7
(\"\"$!\"*\"#7F-\"\"'\"#:7(F,!\"(\"\")!\"&F3\"\"*7(\"\"!F,!\"'F/\"\"%F
4%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "RowOperation(
%, [2,1], -1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")'*)>N
\"-%'MATRIXG6#7%7(\"\"$!\"*\"#7F-\"\"'\"#:7(\"\"!\"\"#!\"%\"\"%F3!\"'7
(F2F,F6F/F5!\"&%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 
"RowOperation(%, [3,2], -3/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'R
TABLEG6%\")7)*y6-%'MATRIXG6#7%7(\"\"$!\"*\"#7F-\"\"'\"#:7(\"\"!\"\"#!
\"%\"\"%F3!\"'7(F2F2F2F2\"\"\"F5%'MatrixG" }}}{EXCHG {PARA 259 "" 0 "
" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 265 10 "Example 2b" }{TEXT 
-1 33 "- obtain the reduced echelon form" }}{PARA 0 "" 0 "" {TEXT -1 
79 "Now that B has been put into an echelon form, put it into reduced \+
echelon form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "RowOperati
on(%, [1,3], -6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")k$
*)>\"-%'MATRIXG6#7%7(\"\"$!\"*\"#7F-\"\"!F-7(F/\"\"#!\"%\"\"%F1!\"'7(F
/F/F/F/\"\"\"F3%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"RowOperation(%, [2,3], -2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTA
BLEG6%\")Sc7:-%'MATRIXG6#7%7(\"\"$!\"*\"#7F-\"\"!F-7(F/\"\"#!\"%\"\"%F
/!#97(F/F/F/F/\"\"\"F3%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "RowOperation(%, 2, 0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
'RTABLEG6%\"(_K4$-%'MATRIXG6#7%7(\"\"$!\"*\"#7F-\"\"!F-7($F/F/$\"#5!\"
\"$!#?F4$\"#?F4F1$!#qF47(F/F/F/F/\"\"\"\"\"%%'MatrixG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "RowOperation(%, [1,2], 9);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")#pc>\"-%'MATRIXG6#7%7($
\"\"$\"\"!$F.F.$!#g!\"\"$\"#!*F2F/$!$?(F27(F/$\"#5F2$!#?F2$\"#?F2F/$!#
qF27(F.F.F.F.\"\"\"\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "RowOperation(%, 1, 1/3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\")'4)[8-%'MATRIXG6#7%7($\"+++++5!\"*$\"\"!
F0$!+++++?F.$\"+++++IF.F/$!+++++C!\")7(F/$\"#5!\"\"$!#?F;$\"#?F;F/$!#q
F;7(F0F0F0F0\"\"\"\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
38 "This is the reduced echelon form of B." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Here's the \+
\"fast\" way:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ReducedRow
EchelonForm(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")W%R@
\"-%'MATRIXG6#7%7(\"\"\"\"\"!!\"#\"\"$F-!#C7(F-F,F.\"\"#F-!\"(7(F-F-F-
F-F,\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "25 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }{RTABLE_HANDLES 13650332 12064356 14907564 14925544 12749376
 12784328 13519896 11789812 11989364 15125640 3093252 11956692 
13488096 12139444 }{RTABLE 
M7R0
I5RTABLE_SAVE/13650332X,%)anythingG6"6"[gl!"%!!!#*"$"$"""""#""$""%""&""'""("")"
"*F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/12064356X,%)anythingG6"6"[gl!"%!!!#*"$"$"#6""#""$"#H""&""'"#Z"")"
"*F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/14907564X,%)anythingG6"6"[gl!"%!!!#*"$"$"""""$""#""%""'""&""(""*"
")F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/14925544X,%)anythingG6"6"[gl!"%!!!#*"$"$"""$"#5!""""$""%$"#DF*""'
""($"#SF*""*F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/12749376X,%)anythingG6"6"[gl!"%!!!#3"$"'""!""$F(F(!"(!"*!"'"")"#7
""'!"&F*""%F,F.F/""*"#:F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/12784328X,%)anythingG6"6"[gl!"%!!!#3"$"'""$F'""!!"*!"(F'"#7"")!"'
F)!"&""'F/F,""%"#:""*F.F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13519896X,%)anythingG6"6"[gl!"%!!!#3"$"'""$""!F(!"*""#F'"#7!"%!"'
F)""%""'F/F*F."#:F-!"&F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/11789812X,%)anythingG6"6"[gl!"%!!!#3"$"'""$""!F(!"*""#F("#7!"%F(F
)""%F(""'F*""""#:!"'F-F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/11989364X,%)anythingG6"6"[gl!"%!!!#3"$"'""$""!F(!"*""#F("#7!"%F(F
)""%F(F(F*"""F)!"'F-F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/15125640X,%)anythingG6"6"[gl!"%!!!#3"$"'""$""!F(!"*""#F("#7!"%F(F
)""%F(F(F("""F)!#9F-F&
}
{RTABLE 
M7R0
I4RTABLE_SAVE/3093252X,%)anythingG6"6"[gl!"%!!!#3"$"'""$$""!F)F)!"*$"#5!""F)"#7
$!#?F-F)F*$"#?F-F)F)F("""F*$!#qF-""%F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/11956692X,%)anythingG6"6"[gl!"%!!!#3"$"'$""$""!$F)F)F)F*$"#5!""F)
$!#gF-$!#?F-F)$"#!*F-$"#?F-F)F*F*"""$!$?(F-$!#qF-""%F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13488096X,%)anythingG6"6"[gl!"%!!!#3"$"'$"+++++5!"*$""!F+F+F*$"#5
!""F+$!+++++?F)$!#?F.F+$"+++++IF)$"#?F.F+F*F*"""$!+++++C!")$!#qF.""%F&
}
{RTABLE 
M7R0
I5RTABLE_SAVE/12139444X,%)anythingG6"6"[gl!"%!!!#3"$"'"""""!F(F(F'F(!"#F)F(""$"
"#F(F(F(F'!#C!"(""%F&
}