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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 204 29 "Determinants by Row Redu
ction" }}{PARA 0 "" 0 "" {TEXT 205 0 "" }}{PARA 0 "" 0 "" {TEXT 205 
35 "Worksheet written by Russell Blyth." }}{PARA 0 "" 0 "" {TEXT 205 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "restart: with(Linear
Algebra):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 206 7 "Outline" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT 200 65 "Use Maple to compute determinants using \+
elementary row operations" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "
Typesetting:-mo(\"⁢\", form = \"\", fence = \"false\", \+
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 {PARA 3 "" 0 "" {TEXT 206 48 "Utilizing row operations to compute det
erminants" }}{EXCHG {PARA 0 "" 0 "" {TEXT 205 145 "We compute the dete
rminant of a matrix using elementary row operations. We need to keep t
rack of the effect of each row operation as we go along." }}{PARA 0 ""
 0 "" {TEXT 205 0 "" }}{PARA 0 "" 0 "" {TEXT 205 83 "We generate a ran
dom 6x6 matrix, and check that it has full rank before proceeding." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A := RandomMatrix(6,6,gener
ator=rand(-5..5));\n" }{MPLTEXT 1 0 1 "R" }{MPLTEXT 1 0 7 "ank(A);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 205 203 "We reduce A to row echelon form
, one step at a time. Start with the first column. For the row operati
ons we note specifically that adding a multiple of one row to another \+
does not affect the determinant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 54 "mf := -A[2,1]/A[1,1]:
B := RowOperation(A,[2,1],mf):\n" }
{MPLTEXT 1 0 22 "mf := -B[3,1]/B[1,1]:
" }{MPLTEXT 1 0 32 "B := RowOpe
ration(B,[3,1],mf):\n" }{MPLTEXT 1 0 22 "mf := -B[4,1]/B[1,1]:
" }
{MPLTEXT 1 0 32 "B := RowOperation(B,[4,1],mf):\n" }{MPLTEXT 1 0 22 "m
f := -B[5,1]/B[1,1]:
" }{MPLTEXT 1 0 32 "B := RowOperation(B,[5,1],mf)
:\n" }{MPLTEXT 1 0 22 "mf := -B[6,1]/B[1,1]:
" }{MPLTEXT 1 0 30 "B := \+
RowOperation(B,[6,1],mf);" }{MPLTEXT 1 0 2 "\n" }}}{EXCHG {PARA 0 "" 0
 "" {TEXT 205 35 "Next move on the the second column:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "mf := -B[3,2]/B[2,2]:
B := RowOpera
tion(B,[3,2],mf):\n" }{MPLTEXT 1 0 22 "mf := -B[4,2]/B[2,2]:
" }
{MPLTEXT 1 0 32 "B := RowOperation(B,[4,2],mf):\n" }{MPLTEXT 1 0 22 "m
f := -B[5,2]/B[2,2]:
" }{MPLTEXT 1 0 32 "B := RowOperation(B,[5,2],mf)
:\n" }{MPLTEXT 1 0 22 "mf := -B[6,2]/B[2,2]:
" }{MPLTEXT 1 0 30 "B := \+
RowOperation(B,[6,2],mf);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 17 "The
 third column:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "mf := -B[
4,3]/B[3,3]:
B := RowOperation(B,[4,3],mf):\n" }{MPLTEXT 1 0 22 "mf :=
 -B[5,3]/B[3,3]:
" }{MPLTEXT 1 0 32 "B := RowOperation(B,[5,3],mf):\n"
 }{MPLTEXT 1 0 22 "mf := -B[6,3]/B[3,3]:
" }{MPLTEXT 1 0 30 "B := RowO
peration(B,[6,3],mf);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 14 "Fourth \+
column:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "mf := -B[5,4]/B[
4,4]:
B := RowOperation(B,[5,4],mf):\n" }{MPLTEXT 1 0 22 "mf := -B[6,4
]/B[4,4]:
" }{MPLTEXT 1 0 30 "B := RowOperation(B,[6,4],mf);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 205 13 "Fifth column:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 52 "mf := -B[6,5]/B[5,5]:
B := RowOperation(B,[
6,5],mf);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 107 "Now the determinan
t is the product of the diagonal entries. We compute the determinant d
irectly as a check." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "B[1,
1]*B[2,2]*B[3,3]*B[4,4]*B[5,5]*B[6,6];\n" }{MPLTEXT 1 0 15 "Determinan
t(A);" }}}{SECT 1 {PARA 209 "" 0 "" {TEXT 200 8 "Exercise" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT 200 245 "1)  Generate a random 60 x 60 matrix an
d time how long it takes to compute the determinant; compare this time
 with the time required for a 61x61 matrix. Based on your observations
, what method do you think Maple is using to compute determinants?" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "st := time():A60 := RandomM
atrix(60,60,generator=rand(-5..5)):Determinant(A60);time() - st;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "st := time():A61 := RandomMa
trix(61,61,generator=rand(-5..5)):Determinant(A61);time() - st;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 200 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}
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