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Unformatted text preview: Fok Am. FOL Airs an an
ouzA= iwu‘lnQuQ“ MA—v Q“ en' Q“ , Q“ QLIQ11+ qr], Q1$Q1‘*Q\3 Qt‘Qn
" quuq's" Q uqnqu' Q'uq nQg; an 01.1, Dr. Doom Lin Alg MATHl850/2050 Section 2.1 Page 1 of 6 Chapter 2  Determinants
Section 2.1  Determinants by Cofactor Expansion Think of determinants as a function that takes as input a square matrix,
and produces as output a real number. In this section we calculate determi nants in a more practical way than from the original deﬁnition of Section
2.4. Deﬁnition If A is a square matrix, then the minor of entry am is denoted
by Mi] and is deﬁned to be the determinant of the submatrix that remains
after the ith row and jth column are deleted from A. The number (—1)i+~7Mzj
is denoted by Ci] and it is called the cofactor of entry aij. . . . K6 Foo. 315
Example: Finding Minors and Cofactors S m" QM" o
1 0 ‘1 1 M = 2\I><Z; 0/:
2 0 1 2 l\ —\/z\<z —\\7.
A:
0 0 1 0
i1 _1 2 2J = 0+ 01'0 ’(—\l1+O+O)= 2/ Cu = 60‘“ M“: 1/, 1 (+1
0 " Li'l'sl/l CuteI) ”11.1 ll
7. Dr. Doom Lin Alg MATH1850/2050 Section 2.1 Page 2 of 6 Theorem 2.1.1. Expansions by Cofactors The determinant of an n X 71 matrix A can be computed by multiplying the
entries of any row (or column) by their respective cofactors, and then adding
up those products. That is d6t<A> = @101ij CLQJ'OQJ' ‘i— . . . + CLW'CW'
(expansion along the j hcolumn) and
d6t<A> = (@1021 + @2022 ‘i— . . . + amom (expansion along the z' h.row) Note: We may choose any row or column for expansion, so for efﬁciency, it
makes sense to choose a row or column with “nice” entries  that is with lots of 0’s. Exam 1e: Cofactor Ex ansion
p p Amur. ‘tHE‘ '5“ [Iowa OIL cocacmn. EacMMsle AUmL A“ Mo.) 0
MA = a uC n J‘ qth/CL ‘kQBCBA— Qmcw
m o l 7 H; 1 "
=\.(—\) \Olo\+QI)L—\\ (”Irib) Do \"L'L ., o+o‘_(\)é‘)m\i :\ + O ‘ V60“; ' ‘ 4 Dr. Doom Lin Alg MATHl850/2050 Section 2.1 Page 3 of 6 Deﬁnition If A is an n X n matrix and Ci] is the cofactor of aij then the
matrix Cm C772 Cm is called the cofactor matrix of A. The transpose of the cofactor matrix is
called the adjoint of A and is denoted by adj(A). 0 l
1 2 —2 C..=(—i)”‘ H _s = —q.
Example: WithA= —1 0 1
\‘VL _\ ‘
2 4 —5 Cu: (q \= —5,
L s C= 2 —\ o CT=wRA=[EI'll 0 7.. Dr. Doom Lin Alg MATH1850/2050 Section 2.1 Page 4 of 6 The importance of the next formula does not come from its value as a com
putational tool (the method from Chapter 1 that uses rowreduction is much
more efﬁcient) but rather from its uses in proving properties of the inverse. Theorem 2.1.2. Inverse of a Matrix Using Its Adjoint
If A is an invertible matrix, then 1 A— 1_
_det(A) adj (A). Implicit in the identity above is that if A is invertible, then det(A) 7é 0; we
shall prove in Section 2.3 that in fact, A is invertible if and only if det(A) # 0. The proof uses the following Lemma: Lemma If A is any n X 71 matrix, then for any 2' 55 j we have
(@1011 + CLZQOJ'Q + . . . (Lin/OJ” = 0. In words, if we multiply the entries of one row by the corresponding cofactors
of a different row and then sum up those products we will obtain 0. MPH‘O
Proof of 171771.212: ‘A «RA = MA‘IV. 1&3 H“ A ”($19: a Tatsz“ is whkf we Show (A'MAB‘S = (AGX, —. aﬁ‘ci‘IQi1C51*+Q‘IKC3V\
— 0 ‘Q ('15 (FILM LEMMA)
— EMA IQ 6:3
=(MATM);S . bone. Dr. Doom Lin Alg MATH1850/2050 Section 2.1 Page 5 of 6 Theorem 1.7.1.(d) Properties of Triangular Matrices
The inverse of a lower triangular matrix is lower triangular, and the inverse
of an upper triangular matrix is upper triangular. Proof: U$IL18 A = :l‘lt—AQJJSA. WE'LL SHOU THE Mace Fol. A WA NT Mt; is MT. 8F UHEL A M'ralx um“ A ZGILO «A bmaouAgrnﬁs Cs 2am. Example: Using the Adjoint to ﬁnd an Inverse 1 2 —2 —4 2 2
WithAz —1 0 1 we got adj(A)= —3 —1 1
2 4 —5 —4 0 2
o
CLtA: QHCUAoyZLl—QHC” ' G\)(_31+\ \‘lLt '7;\ + \(qYhs \\ :\
: “1/.937 0 Dr. Doom Lin Alg MATH1850/2050 Section 2.1 Page 6 of 6 Theorem 2.1.4. Cramer’s Rule If Ax = b is a linear system of n equations
in n unknowns, and if det(A) 7é 0 then the system has a unique solution. The
solution is _ det(A1) _ det(A2) _ d675<14n> _ det(A)’ 562— det(A)’ “7””— det(A)’ 331 where the matrix 143 (j = 1, 2, . . . ,n) is obtained from the matrix A by re
placing its jth column by the column of constants b. Example:
171  2:62 — 2363 = 2 _ l 7 7.
—331 + 333 = —2 A— ’\ o '
2331—4332—5333: 4 1 “ ‘5
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= MAI _ "Ll _
X‘ MK ' '7. '9‘//'
l 'L "L _ MA» ” Q— _
Az'X'I z \ “(A1. ‘0 6° XV MA 'L '0
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 Spring '09
 MihaiBeligan
 Linear Algebra, Algebra

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