Determinants Chapter 3


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1 Determinnts hpter
2 Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse! more lter Exmple: 7
3 Generliztion to nxn Mtrix Definition Let e squre n x n mtrix. Let M ij denote the (n)x(n) squre sumtrix of otined y deleting its ith row nd jth column. The determinnt M ij is clled the (i,j) minor of nd we define the (i,j) cofctor of s the signed minor ij = () i+j M ij Exmple : ) M 7 8 ( ) M ( ) ) M nd
4 lcultion Of Determinnt Lplce Formul : The determinnt of squre mtrix = [ ij ] is given y n j ij ij Expnd out ith row n i ij ij Expnd out jth column order recursive! E ercise ppl this form l to clc lte the determinnt of the mtri Exercise : pply this formul to clculte the determinnt of the x mtrix on previous slide y expnding over row nd over column
5 Properties of Determinnts ) = T Proof omitted; give x nd x exmples ) Let e squre mtrix: (cn e shown y pplying Lplce s formul) (i) If hs row (column) of zeros = Φ (ii) If hs identicl rows (columns) = Φ (iii) If is tringulr, = product of digonl elements (iv) If is digonl = product of digonl elements
6 Properties of Determinnts ) Suppose B is otined from y elementry row (column) opertions, then: i. If rows (columns) were interchnged B =  ution ii. If row (column) of ws multiplied y sclr B = iii. If multiple of row (column) ws dded to nother row (column) then B = Exercise : verify these properties p for generl x mtrix ution det( B ) det( ) det( B ) ) det(b) = det() det(b) : generlizes to product of N mtrices Specil cse: det( ) = [det()] det( n ) = [det()] n
7 Properties of Determinnts ) Let e squre n x n mtrix. The following sttements re equivlent: (i) is invertile (ii) x = hs only the zero solution (iii) x= hs unique solution (iv) hs n nonzero pivots (v) det() = : follows y pplying ppy elementry opertions (vi) rn () = n ) det(α) = α n det() : follows from repeted ppliction of ()(ii) n times 7)  = / (since  = I) 8) = Product of pivots (nother wy to clculte determinnt)
8 Simplified lcultion Of Determinnt lgorithm ) hoose n element positions ij s pivot ) pply elementry opertions to put 's in ll the other in the columns (row) contining ) Use Lplce Formul to expnd the determinnt y the column (row) contining ij ij
9 Exmple (Introduce zeros in the third column) expnd out rd column ( ) 7 () 87 
10 More Definitions! djoint of Mtrix Definition : dj [ ] Exmple : (mtrix of ofctors trnsposed) [ ] 8 ; ; ij T ; Exercise : lculte nd.dj() ; ; ; 8 dj[ ] 8 ; ; 8;
11 ppliction : Mtrix Inversion Using Determinnts Theorem : ( dj( )) ( dj( )) I (cn e used to find (i, dj( ) [ ] ij Exmple : From the previous exmple det( ), j) element of T  ) Note : This formul will sve you time if you wnt to compute only or few elements of the inverse since dj( ) ( i, j, i / j) ( dj( )( i, 8 8 j)) /
12 ppliction : solving x= using determinnts determinnts : equtions x system of cse of specil Strt with Y Y Y Y Hence ; Y
13 emrs )Determinnt of coefficient mtrix ppers s the denomintor of oth quotients )Numertor for nd Y is otined y sustituting the column of constnts t in plce of the column of coefficients of the corresponding unnown )System hs unique solution iff =
14 Generliztion :rmer s ule unique hs x equtions liner of system squre The : Theorem " " y of column ith the replcing y from otined is where / y x given is solution the nd iff solution unique hs x equtions liner of system squre The : Theorem i i i
15 E l S l i l Exmple: Z Y Z Y Solve using rmer s rule Z Y Z Y ; ; Y Z
16 ppliction : ross Product u v ppliction : Torque
17 eview (hpter ) Q : Determine K so tht the following system hs (i) Unique Solution, (ii) No Solution, (iii) Infinite Solutions (Prmeterize them) Z Y Y Z Y Z Y
18 eview (hpter ) (i) Unique Solution non   zero pivots Z Y Z Y (ii) rd Eqution : no solution (iii)  rd Eqution : solutions  Y Y Z  Z  Y  Z  y  Z (Z is free vrile)
19 eview (hpter ) Q: True or Flse: Justify if True or give proof (Exmple not enough!) nd correct or give counter exmple if Flse ) ) ll If upper tri nd (B) Exmple : T B re B ngulr T mtrices symmetric, T B then B re in echelon B is form symmetric not symmetric F F
20 hpter eview Prolem ompute & :
21 Solve y determinnts x y x y hpter eview x 78 y
22 eview (hpter ) eview (hpter ) : Flse or True digonl is dj() I.dj() digonl is dj() then digonl is If (i) digonl digonl dj()  digonl digonl digonl digonl Fl dj ii n ) ( ) digonl B B If iii dj dj Flse n n ) ) ( ) ( B B If iii )
23 eview (hpter ) Solve Using rmer s ule
24 eview (hpter ) eview (hpter ) 7 9 ; 7 9 ; 8; 7 9 ;
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