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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': vimGUM) 131") 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‘ Anamen. is
a‘lLQKO‘ZHQRI 6153 = 161)},{451 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 ~ . . . ~ amn multiplied by the
sign of the associated permutation (jbjg, . . . ,jn).
@ 2 1 —1 —2 mu m slmo
—1 0 1 @ —2 Signen En‘u mowers.
Example: With A = 0 1 Q) 2 0 ,
—2 Q) —3 5 3
—1 1 2 0 @
Quatuqsaauras; = 1091605‘ =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) = manmars. 0/91 MCQ‘ 1.361% Q.\o + Mic—7.) — (Ls0+ 2lL—1H (MC1)) = — 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\=+<(.\)igCvt)3)—(601511) = '7
@ at c; <02>GD ('L,H,\,S,‘5) (L.S',\,H,">) o 0 ® even 0m S/f ...
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 Spring '09
 MihaiBeligan
 Linear Algebra, Algebra

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