2.1 - Fok A-m FOL Airs an an ouzA= iwu‘ln-QuQ“ MA—v...

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Unformatted text preview: Fok A-m. FOL Airs an an ouzA= iwu‘ln-QuQ“ MA—v Q“ en' Q“ , Q“ QLIQ11+ qr], Q1$Q1‘*Q\3 Qt‘Qn " quuq's" Q- uqnqu' Q'uq nQ-g; an 0-1.1, 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+~7Mz-j 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 ’(—\-l-1+O+O)= 2/ Cu -= 60‘“ M“: 1/,- 1 (+1 0 " Li'l'sl/l Cute-I) ”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 ‘k-QBCBA— Q-mcw m o l 7- H; 1 " =\.(—\) \Olo\+QI)L—\\ (”I-rib) 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 row-reduction 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 ”($19: a Tats-z“ is whkf we Show (A'MAB‘S = (AG-X, —. afi‘ci‘IQi1C51*---+Q‘IKC3V\ — 0 ‘Q ('15 (FILM LEMMA) — EMA IQ 6:3 =(MA-TM);S . bone. Dr. Doom Lin Alg MATH1850/2050 Section 2.1 Page 5 of 6 Theorem 1.7.1.(d) Properties of Triangular Matrices The inverse of a lower triangular matrix is lower triangular, and the inverse of an upper triangular matrix is upper triangular. Proof: U$IL18 A = :l-‘lt—A-QJJSA. 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: QHCUA-oyZL-l—QHC” ' G\)(_31+\ \‘lL-t '7;\ + \(qY-hs \\ :\ : “1/.937 0 Dr. Doom Lin Alg MATH1850/2050 Section 2.1 Page 6 of 6 Theorem 2.1.4. Cramer’s Rule If Ax = b is a linear system of n equations in n unknowns, and if det(A) 7é 0 then the system has a unique solution. The solution is _ det(A1) _ det(A2) _ d675<14n> _ det(A)’ 562— det(A)’ “7””— det(A)’ 331 where the matrix 143- (j = 1, 2, . . . ,n) is obtained from the matrix A by re- placing its jth column by the column of constants b. Example: 171 -- 2:62 — 2363 = 2 _ l 7- -7. —331 + 333 = —2 A— ’\ o ' 2331—4332—5333: 4 1 “ ‘5 424sz Go so 62W: 'L '1. ‘1 _ KULE 6009.145) A“ —7. o | “(A'lzC-Zwllw‘ Z l +l-G\)"‘"’ 7' 1 Li H —S 3;: CH4 : vq —-L 0 = MAI _ "Ll _ X‘ MK ' '7. '9‘//' l 'L "L _ MA» ” Q— _ Az'X'I -z |\ “(A1. ‘0 6° XV MA '-L '0 Z R “S ...
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