# 2.4 - Dr Doom Lin Alg MATH1850/2050 Section 2.4 Page 1 of 4...

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Unformatted text preview: Dr. Doom Lin Alg MATH1850/2050 Section 2.4 Page 1 of 4 Chapter 2 - Determinants Section 2.4 - A Combinatorial Approach to Determinants This section contains the deﬁnition of determinants. For Whatever reason, the book’s author chose to end the chapter on determinants with it. It’s a short section, and we’ll do things in a slightly different order. Deﬁnition A permutation of the set of integers {1,2, . . . ,n} is an ar- rangement of these integers in some order Without omissions or repetitions. Example: Permutations on 3 Integers {M153 3?. 3'. .L 50 Time ARE gun—.3! =9 vevnummm 0N 3 oqws. ()3 GEUEML ‘TﬁEIE Avg M. on vx 08566.15. (nl': vim-GUM) 1-31") 012,33 (MEI) (1“)3) (2)1’)\) (3,1,1) (\$.13) Deﬁnition An inversion is said to occur in a permutation (j1,j2, . . . ,jn) on the ﬁrst n positive integers, Whenever a larger integer precedes a smaller one. Example: Counting Inversions (a) (5737177747276) ' ‘ (b) (67275747173) H"UO*3+\+O = \%\906u\ous S*\‘\'5*2‘\'O _ u \u‘vegslons_ Dr. Doom Lin Alg MATHl850/2050 Section 2.4 Page 2 of 4 Deﬁnition A permutation if called even if the total number of inversions is an even integer, and is called odd if the total number of inversions is odd. Example: Permutations on {1, 2,3} Permutation Number of lnversions Classiﬁcation (17 2, 3) 0 lave») (17 3, 2) ‘ °°° (2, 1, 3) l °°° (2,3,1) 9- W (3,1,2) 1 We” (3,2,1) 3 °°“ Deﬁnition By an elementary product from an n X 71 matrix A we shall mean any product of n entries from A, no two of which come from the same row or the same column. Example: With A 2 “were ME S\_ =\zo Emmemuw mucous FOR. A. a»; sum GU’H mower Ls aquzxqz's O‘QSQS‘L = (“\)(rl)'2,-'6-\= él‘ Ana-men. is a‘lLQKO‘ZHQRI 6153 = 16-1)},{45-1 5 31-": Dr. Doom Lin Alg MATH1850/2050 Section 2.4 Page 3 of 4 Deﬁnition The Sign of a permutation, is +1 if the permutation is even, and —1 if the permutation is odd. By a Signed elementary product from A we shall mean an elementary product aljlagjz ~ . . . ~ am-n multiplied by the sign of the associated permutation (jbjg, . . . ,jn). @ 2 1 —1 —2 mu m slmo —1 0 1 @ —2 Sign-en En‘u mowers. Example: With A = 0 1 Q) 2 0 , —2 Q) —3 5 3 —1 1 2 0 @ Quatuqsaauras; = 1091-60-5‘ =é0- \'< AM ELT‘RY 92°60“, {T's “Sm-""5” we“ ‘5 (514,752.13) mum“ is out L'smilmsulmc) , So swim) gulp, ewe? is —— 40. Example: Say A is a 2 X 2 matrix. Signed Elementary Associated Even or Elementary Product Permutation Odd Product allagg (1, 2) €V€Il +CL11€L22 a12a21 (27 1) Odd —CL12CL21 Say A is a 3 X 3 matrix. Signed Elementary Associated Even or Elementary Product Permutation Odd Product a11a22a33 (l , 1, '5) EVEN Q“Q~,_-,,Q33 a11a23a32 ( i ,5, 7,) 0M - 0.624,;sz a12a21a33 C 7., \ , 5) 0 DB “04792.st a12a23a31 (7. ,1, , l) even «10.15am a13a21a32 C 7; , \,1) EVW Q—qau Q1; a13a22a31 C “5,1..\) om — qnqna,‘ Dr. Doom Lin Alg MATHl850/2050 Section 2.4 Page 4 of 4 Deﬁnition Let A be a square matrix. We deﬁne det(A) to be the sum of all signed elementary products from A. Examples: From the tables above we see that: (a) If A is a 2 X 2 matrix, then d6t<A> = allagg — algagl. (b) If A is a square matrix of order 3, then d675<14> = a11a22a33 + a12a23a31 + a13a21a32 — a13a22a31 — a12a21a33 — a11a23a32- 'L K O I/l WORKS ONLI FOR. \ >< - *2 3x | 2 1 0 1 2 3 1} . _ 1 2 _ \ / _ 0 5 3 —2 Examples. WlthA— [3&2 ] 7B_ 2X?X:§ 70_ 0 [email protected] 1 N0 L0 0 0 @j (\A\ =) AMA) = man-mars. 0/91 MCQ‘ 1.361% Q.\-o + Mic—7.) —- (Ls-0+ 2-l-L—1H (MC-1)) = — lO/_ [email protected])=+A'S 'l'l “- (O (“NEE ME. 2% Stine» Mammy mamas, \$07 I u, op Tm Au am). 0 @ o o o b: 0 ° 04:? &,1(b\=+<(.\)i-g-Cvt)-3)—(60-15-11) = '7 @ at c; <02>GD ('L,H,\,S,‘5) (L.S',\,H,">) o 0 ® even 0m S/f ...
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