๐๐๐ญ๐๐ซ๐ฆ๐ข๐ง๐๐ง๐ญ ๐จ๐ ๐ ๐๐๐ญ๐ซ๐ข๐ฑ
(can calculate for square matrices up to an order of 5)
18
Expression 19: "D" Subscript, "e" "t" , Baseline left parenthesis, "L" , right parenthesis equals left brace, "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket equals "L" left bracket, length left parenthesis, "L" , right parenthesis minus 1 , right bracket : left brace, "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket equals 1 : "L" left bracket, 1 , right bracket , "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket equals 2 : "D" Subscript, "e" "t" 2 , Baseline left parenthesis, "L" , left bracket, 1...4 , right bracket , right parenthesis , "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket equals 3 : "D" Subscript, "e" "t" 3 , Baseline left parenthesis, "L" , left bracket, 1...9 , right bracket , right parenthesis , "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket equals 4 : "D" Subscript, "e" "t" 4 , Baseline left parenthesis, "L" , left bracket, 1...1 6 , right bracket , right parenthesis , "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket equals 5 : "D" Subscript, "e" "t" 5 , Baseline left parenthesis, "L" , left bracket, 1...2 5 , right bracket , right parenthesis , right brace , right braceDetโL=LlengthL=LlengthLโ1:LlengthL=1:L1,LlengthL=2:Det2โL,1...4,LlengthL=3:Det3โL,1...9,LlengthL=4:Det4โL,1...16,LlengthL=5:Det5โL,1...25
19
๐๐๐ญ๐ซ๐ข๐ฑ ๐จ๐ ๐๐ข๐ง๐จ๐ซ๐ฌ
(can calculate for square matrices up to an order of 6)
20
Expression 21: "M" Subscript, "m" "o" "m" , Baseline left parenthesis, "L" , right parenthesis equals left brace, "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket equals "L" left bracket, length left parenthesis, "L" , right parenthesis minus 1 , right bracket : Start sum from "n" Subscript, 101 , Baseline equals left bracket, 1...length left parenthesis, "L" , right parenthesis , right bracket to left bracket, 1...length left parenthesis, "L" , right parenthesis , right bracket , end sum, left brace, "n" Subscript, 101 , Baseline less than or equal to length left parenthesis, "L" , right parenthesis minus 2 : "D" Subscript, "e" "t" , Baseline left parenthesis, "E" Subscript, "x" "t" "r" "a" "c" "t" 2 , Baseline left parenthesis, "L" , "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket , "R" Subscript, "o" "w" 1 , Baseline left parenthesis, "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket , "n" Subscript, 101 , Baseline , right parenthesis , "C" Subscript, "o" "l" "u" "m" "n" , Baseline left parenthesis, "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket , "n" Subscript, 101 , Baseline , right parenthesis , right parenthesis , right parenthesis , "L" left bracket, "n" Subscript, 101 , Baseline , right bracket , right brace , right braceMmomโL=LlengthL=LlengthLโ1:1...lengthLโn101โ=1...lengthLn101โโคlengthLโ2:DetโExtract2โL,LlengthL,Row1โLlengthL,n101โ,ColumnโLlengthL,n101โ,Ln101โ
21
๐๐๐ญ๐ซ๐ข๐ฑ ๐จ๐ ๐๐จ๐๐๐๐ญ๐จ๐ซ๐ฌ
(can calculate for square matrices up to an order of 6)
22
Expression 23: "M" Subscript, "m" "o" "c" , Baseline left parenthesis, "L" , right parenthesis equals left brace, "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket equals "L" left bracket, length left parenthesis, "L" , right parenthesis minus 1 , right bracket : Start sum from "n" Subscript, 1 , Baseline equals left bracket, 1...length left parenthesis, "L" , right parenthesis , right bracket to left bracket, 1...length left parenthesis, "L" , right parenthesis , right bracket , end sum, left brace, "n" Subscript, 1 , Baseline less than or equal to length left parenthesis, "L" , right parenthesis minus 2 : left parenthesis, negative 1 , right parenthesis Superscript, left parenthesis, "R" Subscript, "o" "w" 1 , Baseline left parenthesis, "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket , "n" Subscript, 1 , Baseline , right parenthesis plus "C" Subscript, "o" "l" "u" "m" "n" , Baseline left parenthesis, "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket , "n" Subscript, 1 , Baseline , right parenthesis , right parenthesis , Baseline "M" Subscript, "m" "o" "m" , Baseline left parenthesis, "L" , right parenthesis left bracket, "n" Subscript, 1 , Baseline , right bracket , "L" left bracket, "n" Subscript, 1 , Baseline , right bracket , right brace , right braceMmocโL=LlengthL=LlengthLโ1:1...lengthLโn1โ=1...lengthLn1โโคlengthLโ2:โ1Row1โLlengthL,n1โ+ColumnโLlengthL,n1โMmomโLn1โ,Ln1โ
23
๐๐ง๐ฏ๐๐ซ๐ฌ๐ ๐จ๐ ๐ ๐๐๐ญ๐ซ๐ข๐ฑ
(can calculate for square matrices up to an order of 5)
24
Expression 25: "M" Subscript, "i" "n" "v" , Baseline left parenthesis, "L" , right parenthesis equals Start sum from "n" Subscript, 20 , Baseline equals left bracket, 1...length left parenthesis, "L" , right parenthesis , right bracket to left bracket, 1...length left parenthesis, "L" , right parenthesis , right bracket , end sum, left brace, "n" Subscript, 20 , Baseline less than or equal to length left parenthesis, "L" , right parenthesis minus 2 : StartFraction, 1 Over "D" Subscript, "e" "t" , Baseline left parenthesis, "L" , right parenthesis , EndFraction times "T" left parenthesis, "M" Subscript, "m" "o" "c" , Baseline left parenthesis, "L" , right parenthesis , right parenthesis left bracket, "n" Subscript, 20 , Baseline , right bracket , "L" left bracket, "n" Subscript, 20 , Baseline , right bracket , right braceMinvโL=1...lengthLโn20โ=1...lengthLn20โโคlengthLโ2:1DetโLโยทTMmocโLn20โ,Ln20โ
Expression 27: "T" Subscript, "r" "u" "e" , Baseline left parenthesis, "L" , right parenthesis equals left brace, length left parenthesis, "L" , right parenthesis minus 2 equals "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket times "L" left bracket, length left parenthesis, "L" , right parenthesis minus 1 , right bracket , right braceTrueโL=lengthLโ2=LlengthLยทLlengthLโ1
27
Expression 28: "D" Subscript, "o" "t" , Baseline left parenthesis, "L" , "L" Subscript, 2 , Baseline , "R" Subscript, "o" "w" , Baseline , "C" Subscript, "o" "l" , Baseline , right parenthesis equals Start sum from "k" equals 1 to "L" left bracket, length left parenthesis, "L" , right parenthesis , right bracket , end sum, left parenthesis, "I" Subscript, "n" "d" "e" "x" , Baseline left parenthesis, "L" , "R" Subscript, "o" "w" , Baseline , "k" , right parenthesis times "I" Subscript, "n" "d" "e" "x" , Baseline left parenthesis, "L" Subscript, 2 , Baseline , "k" , "C" Subscript, "o" "l" , Baseline , right parenthesis , right parenthesisDotโL,L2โ,Rowโ,Colโ=LlengthLโk=1IndexโL,Rowโ,kยทIndexโL2โ,k,Colโ
28
Expression 29: "C" Subscript, "o" "l" "u" "m" "n" , Baseline left parenthesis, "L" Subscript, 0 , Baseline , "N" , right parenthesis equals left parenthesis, mod left parenthesis, "N" minus 1 , "L" Subscript, 0 , Baseline , right parenthesis plus 1 , right parenthesisColumnโL0โ,N=modNโ1,L0โ+1
29
Expression 30: "R" Subscript, "o" "w" 1 , Baseline left parenthesis, "L" Subscript, 0 , Baseline , "N" , right parenthesis equals left parenthesis, StartFraction, left parenthesis, "N" minus left parenthesis, mod left parenthesis, "N" minus 1 , "L" Subscript, 0 , Baseline , right parenthesis plus 1 , right parenthesis , right parenthesis Over "L" Subscript, 0 , Baseline , EndFraction plus 1 , right parenthesisRow1โL0โ,N=NโmodNโ1,L0โ+1L0โโ+1
30
Expression 31: "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, "S" Subscript, "i" , Baseline , "N" Subscript, 1 , Baseline , right parenthesis equals left bracket, "S" Subscript, "i" , Baseline plus 1... "S" squared , right bracket left bracket, left brace, "C" Subscript, "o" "l" "u" "m" "n" , Baseline left parenthesis, "S" Subscript, "i" , Baseline , left bracket, "S" Subscript, "i" , Baseline plus 1... "S" squared , right bracket , right parenthesis equals "N" Subscript, 1 , Baseline : StartFraction, 1 Over 0 , EndFraction , left bracket, "S" Subscript, "i" , Baseline plus 1... "S" squared , right bracket , right brace equals left brace, "C" Subscript, "o" "l" "u" "m" "n" , Baseline left parenthesis, "S" Subscript, "i" , Baseline , left bracket, "S" Subscript, "i" , Baseline plus 1... "S" squared , right bracket , right parenthesis equals "N" Subscript, 1 , Baseline : 10 to the 10th power , left bracket, "S" Subscript, "i" , Baseline plus 1... "S" squared , right bracket , right brace , right bracketExtractโSiโ,N1โ=Siโ+1...S2iโColumnโSiโ,Siโ+1...S2iโ=N1โ:10โ,Siโ+1...S2iโ=ColumnโSiโ,Siโ+1...S2iโ=N1โ:1010,Siโ+1...S2iโ
31
Expression 32: "E" Subscript, "x" "t" "r" "a" "c" "t" 2 , Baseline left parenthesis, "L" , "S" Subscript, "i" , Baseline , "N" Subscript, "r" "o" "w" , Baseline , "N" Subscript, "c" "o" "l" , Baseline , right parenthesis equals left parenthesis, Start sum from "n" Subscript, 111 , Baseline equals left bracket, 1... left parenthesis, "S" Subscript, "i" , Baseline minus 1 , right parenthesis squared plus 2 , right bracket to left bracket, 1... left parenthesis, "S" Subscript, "i" , Baseline minus 1 , right parenthesis squared plus 2 , right bracket , end sum, left brace, "n" Subscript, 111 , Baseline greater than left parenthesis, "S" Subscript, "i" , Baseline minus 1 , right parenthesis squared : "S" Subscript, "i" , Baseline minus 1 , "I" Subscript, "n" "d" "e" "x" , Baseline left parenthesis, "L" , left brace, "R" Subscript, "o" "w" 1 , Baseline left parenthesis, "S" Subscript, "i" , Baseline minus 1 , "n" Subscript, 111 , Baseline , right parenthesis less than "N" Subscript, "r" "o" "w" , Baseline : "R" Subscript, "o" "w" 1 , Baseline left parenthesis, "S" Subscript, "i" , Baseline minus 1 , "n" Subscript, 111 , Baseline , right parenthesis , "R" Subscript, "o" "w" 1 , Baseline left parenthesis, "S" Subscript, "i" , Baseline minus 1 , "n" Subscript, 111 , Baseline , right parenthesis plus 1 , right brace , left brace, "C" Subscript, "o" "l" "u" "m" "n" , Baseline left parenthesis, "S" Subscript, "i" , Baseline minus 1 , "n" Subscript, 111 , Baseline , right parenthesis less than "N" Subscript, "c" "o" "l" , Baseline : "C" Subscript, "o" "l" "u" "m" "n" , Baseline left parenthesis, "S" Subscript, "i" , Baseline minus 1 , "n" Subscript, 111 , Baseline , right parenthesis , "C" Subscript, "o" "l" "u" "m" "n" , Baseline left parenthesis, "S" Subscript, "i" , Baseline minus 1 , "n" Subscript, 111 , Baseline , right parenthesis plus 1 , right brace , right parenthesis , right brace , right parenthesisExtract2โL,Siโ,Nrowโ,Ncolโ=1...Siโโ12+2โn111โ=1...Siโโ12+2n111โ>Siโโ12:Siโโ1,IndexโL,Row1โSiโโ1,n111โ<Nrowโ:Row1โSiโโ1,n111โ,Row1โSiโโ1,n111โ+1,ColumnโSiโโ1,n111โ<Ncolโ:ColumnโSiโโ1,n111โ,ColumnโSiโโ1,n111โ+1
32
Expression 33: "D" Subscript, "e" "t" 2 , Baseline left parenthesis, "L" , "v" , right parenthesis equals "L" left bracket, "v" left bracket, 1 , right bracket , right bracket times "L" left bracket, "v" left bracket, 4 , right bracket , right bracket minus "L" left bracket, "v" left bracket, 2 , right bracket , right bracket times "L" left bracket, "v" left bracket, 3 , right bracket , right bracketDet2โL,v=Lv1ยทLv4โLv2ยทLv3
33
Expression 34: "D" Subscript, "e" "t" 3 , Baseline left parenthesis, "L" , "v" , right parenthesis equals "L" left bracket, "v" left bracket, 1 , right bracket , right bracket "D" Subscript, "e" "t" 2 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 3 , 1 , right parenthesis , right bracket , right parenthesis minus "L" left bracket, "v" left bracket, 2 , right bracket , right bracket "D" Subscript, "e" "t" 2 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 3 , 2 , right parenthesis , right bracket , right parenthesis plus "L" left bracket, "v" left bracket, 3 , right bracket , right bracket "D" Subscript, "e" "t" 2 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 3 , 3 , right parenthesis , right bracket , right parenthesisDet3โL,v=Lv1Det2โL,vExtractโ3,1โLv2Det2โL,vExtractโ3,2+Lv3Det2โL,vExtractโ3,3
34
Expression 35: "D" Subscript, "e" "t" 4 , Baseline left parenthesis, "L" , "v" , right parenthesis equals "L" left bracket, "v" left bracket, 1 , right bracket , right bracket "D" Subscript, "e" "t" 3 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 4 , 1 , right parenthesis , right bracket , right parenthesis minus "L" left bracket, "v" left bracket, 2 , right bracket , right bracket "D" Subscript, "e" "t" 3 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 4 , 2 , right parenthesis , right bracket , right parenthesis plus "L" left bracket, "v" left bracket, 3 , right bracket , right bracket "D" Subscript, "e" "t" 3 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 4 , 3 , right parenthesis , right bracket , right parenthesis minus "L" left bracket, "v" left bracket, 4 , right bracket , right bracket "D" Subscript, "e" "t" 3 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 4 , 4 , right parenthesis , right bracket , right parenthesisDet4โL,v=Lv1Det3โL,vExtractโ4,1โLv2Det3โL,vExtractโ4,2+Lv3Det3โL,vExtractโ4,3โLv4Det3โL,vExtractโ4,4
35
Expression 36: "D" Subscript, "e" "t" 5 , Baseline left parenthesis, "L" , "v" , right parenthesis equals "L" left bracket, "v" left bracket, 1 , right bracket , right bracket "D" Subscript, "e" "t" 4 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 5 , 1 , right parenthesis , right bracket , right parenthesis minus "L" left bracket, "v" left bracket, 2 , right bracket , right bracket "D" Subscript, "e" "t" 4 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 5 , 2 , right parenthesis , right bracket , right parenthesis plus "L" left bracket, "v" left bracket, 3 , right bracket , right bracket "D" Subscript, "e" "t" 4 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 5 , 3 , right parenthesis , right bracket , right parenthesis minus "L" left bracket, "v" left bracket, 4 , right bracket , right bracket "D" Subscript, "e" "t" 4 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 5 , 4 , right parenthesis , right bracket , right parenthesis plus "L" left bracket, "v" left bracket, 5 , right bracket , right bracket "D" Subscript, "e" "t" 4 , Baseline left parenthesis, "L" , "v" left bracket, "E" Subscript, "x" "t" "r" "a" "c" "t" , Baseline left parenthesis, 5 , 5 , right parenthesis , right bracket , right parenthesisDet5โL,v=Lv1Det4โL,vExtractโ5,1โLv2Det4โL,vExtractโ5,2+Lv3Det4โL,vExtractโ5,3โLv4Det4โL,vExtractโ5,4+Lv5Det4โL,vExtractโ5,5
