𝐃𝐞𝐭𝐞𝐫𝐦𝐢𝐧𝐚𝐧𝐭 𝐨𝐟 𝐚 𝐌𝐚𝐭𝐫𝐢𝐱
(can calculate for square matrices up to an order of 5)
18
นิพจน์ 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 braceDetL=LlengthL=LlengthL−1:LlengthL=1:L1,LlengthL=2:Det2L,1...4,LlengthL=3:Det3L,1...9,LlengthL=4:Det4L,1...16,LlengthL=5:Det5L,1...25
19
𝐌𝐚𝐭𝐫𝐢𝐱 𝐨𝐟 𝐌𝐢𝐧𝐨𝐫𝐬
(can calculate for square matrices up to an order of 6)
20
นิพจน์ 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 braceMmomL=LlengthL=LlengthL−1:1...lengthL∑n101=1...lengthLn101≤lengthL−2:DetExtract2L,LlengthL,Row1LlengthL,n101,ColumnLlengthL,n101,Ln101
21
𝐌𝐚𝐭𝐫𝐢𝐱 𝐨𝐟 𝐂𝐨𝐟𝐚𝐜𝐭𝐨𝐫𝐬
(can calculate for square matrices up to an order of 6)
22
นิพจน์ 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 braceMmocL=LlengthL=LlengthL−1:1...lengthL∑n1=1...lengthLn1≤lengthL−2:−1Row1LlengthL,n1+ColumnLlengthL,n1MmomLn1,Ln1
23
𝐈𝐧𝐯𝐞𝐫𝐬𝐞 𝐨𝐟 𝐚 𝐌𝐚𝐭𝐫𝐢𝐱
(can calculate for square matrices up to an order of 5)
24
นิพจน์ 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 braceMinvL=1...lengthL∑n20=1...lengthLn20≤lengthL−2:1DetL·TMmocLn20,Ln20
25
𝐌𝐚𝐢𝐧 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬 𝐚𝐫𝐞 𝐝𝐞𝐟𝐢𝐧𝐞𝐝 𝐮𝐬𝐢𝐧𝐠 𝐭𝐡𝐞 𝐟𝐨𝐥𝐥𝐨𝐰𝐢𝐧𝐠
26
นิพจน์ 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 braceTrueL=lengthL−2=LlengthL·LlengthL−1
27
นิพจน์ 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 parenthesisDotL,L2,Row,Col=LlengthL∑k=1IndexL,Row,k·IndexL2,k,Col
28
นิพจน์ 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 parenthesisColumnL0,N=modN−1,L0+1
29
นิพจน์ 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 parenthesisRow1L0,N=N−modN−1,L0+1L0+1
30
นิพจน์ 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 bracketExtractSi,N1=Si+1...S2iColumnSi,Si+1...S2i=N1:10,Si+1...S2i=ColumnSi,Si+1...S2i=N1:1010,Si+1...S2i
31
นิพจน์ 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 parenthesisExtract2L,Si,Nrow,Ncol=1...Si−12+2∑n111=1...Si−12+2n111>Si−12:Si−1,IndexL,Row1Si−1,n111<Nrow:Row1Si−1,n111,Row1Si−1,n111+1,ColumnSi−1,n111<Ncol:ColumnSi−1,n111,ColumnSi−1,n111+1
32
นิพจน์ 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 bracketDet2L,v=Lv1·Lv4−Lv2·Lv3
33
นิพจน์ 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 parenthesisDet3L,v=Lv1Det2L,vExtract3,1−Lv2Det2L,vExtract3,2+Lv3Det2L,vExtract3,3
34
นิพจน์ 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 parenthesisDet4L,v=Lv1Det3L,vExtract4,1−Lv2Det3L,vExtract4,2+Lv3Det3L,vExtract4,3−Lv4Det3L,vExtract4,4
35
นิพจน์ 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 parenthesisDet5L,v=Lv1Det4L,vExtract5,1−Lv2Det4L,vExtract5,2+Lv3Det4L,vExtract5,3−Lv4Det4L,vExtract5,4+Lv5Det4L,vExtract5,5
