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8.5-one - 8.5 Determinants and Cramer’s Rule 1 Evaluate a...

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Unformatted text preview: 8.5 Determinants and Cramer’s Rule 1. Evaluate a second-order determinant. Deﬁntliun of the Detenninanl at a 2 x 2 Matrix The determinant ofthe- matrix a: bl is denoted by at I" andis cfcfinedby £11 £32 a: hi I}; If]; _T M a: tile Hibl We also saytltatthe value of the second-order determinant a: g: is “ab-3 — ﬂab» “2 ‘ Tn evaluate a secttntl-nrtlﬁr detenutnant. find the dift'ere'ntfe 01‘ the prnduct at the [th diagnnals. “t. ﬁt I fix = inf}: — ti‘gt‘), hi1 50 Duo. :3 +2 2; 2 (7) l h 3 t 1: f3 _ f8 )— +Lcj , 3 :>_ ’2— 3: w (M ) 3 _ aa 2 L 5 7. F :1 .q __ r. 3 I 1 2— ! - e if 1 Ex: " Lf'g: #2 3 9‘1 3 1,—2.2- X: “j” 2. xiii" 2. Solve a system of linear equations in two variables using Cramer’s rule. Solving in Linear System51] Two Variables Using Betcrminants Cranler-‘s Rule If #33: + my cl ﬂax '+' £13.? 1' [‘1 than _ [fl b1 “1 CI ‘61 b: a: [3 xi: and 12m *3: b1 1'1: 531‘. I": '5‘: a: by when: lb 15:: if] H) “3:? ' " In] at?) Liar—team 3x+teyzlg 73 LE 1Q,” , I i ._ .ﬂl :D—kfl-J’ :l A;_:_Db§, 6 8’ 31—} .2159. 3 Li .J). :0 D: “3} I vii-'3 “,L :9; g\ 3% ti 2‘ 3 " X r G 'Irb . 4253’510 \$32)1‘4\354 was; :Dxﬂb LU 1'23 '3 63) i 31 ﬂ L '- --*l 3 ﬂ '5 Lb’ r) 1’ \ c’ 32’ ‘ W (TTVL €3§Nu1i16V13 CKKL Ciao 3. Evaluate a third-order determinant. Deﬁniti-tm (if a Third-ﬁnder Determinant Hg bl um h; .. _ "—' align; + burly} + (Tier-1h}, ~ (13.51161 W {33Cng — \$3.913}. H1. b]. I Definitiun u!“ the Deteminant at“ a 3 it 3 Matrix A third-arder determinant is deﬁncd by :tt‘; ()3 (T; cal. 11; b3 63; I: ‘ 't‘ j]: " I Each :1 an Ill: tight can: Frill Iha ﬁrst atrium. HI bl Cl 1 Evat'uating the Detenninant Ma 3 x 3 Matrix 1. Each of the th rec terms in the definition contains twu factors—a numerical [helm and a scmnd-m‘dcr determinant. 2. The numerical [actor in cash term is :m eminent fmm the first column or the third-urdtlr dctcrminant. 3. Thy: minus sign precedes the: sect-nth term- 4. The semadmrdar determinant that appears in each term is obtained by t:isng nut the mw and the column cuntaining the numerical t‘acttar. h'! ’53 If}; {11 - HI. \$_+113 tr)". {1" :- Tht: minut- m an element is the. determinant that remains afltzt' deleting the. raw and column of that. element. For this rcasmt. we call this method expansiml by minors. ...
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8.5-one - 8.5 Determinants and Cramer’s Rule 1 Evaluate a...

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