Chpter 04.04 Ury trx Opertos After redg ths chpter, you should be ble to:. kow wht ury opertos mes,. fd the trspose of squre mtrx d t s reltoshp to symmetrc mtrces,. fd the trce of mtrx, d 4. fd the ermt of mtrx by the cofctor method. Wht s the trspose of mtrx? Let be m mtrx. he [B] s the trspose of the f b for ll d j. ht s, the th row d the j th colum elemet of s the j th row d th colum elemet of [B]. Note, [B] would be m mtrx. he trspose of s deoted by. Exmple Fd the trspose of 5 0 5 0 6 6 he trspose of s 5 5 0 0 [ A ] 5 5 7 6 6 7 5 7 5 7 Note, the trspose of row vector s colum vector d the trspose of colum vector s row vector. Also, ote tht the trspose of trspose of mtrx s the mtrx tself, tht s, ([ A ] ) [ A]. Also, ( A B) A + B ;( ca) ca +. 04.04.
04.04. Chpter 04.04 Wht s symmetrc mtrx? A squre mtrx [ A ] wth rel elemets where j for,,..., d j,,..., s clled symmetrc mtrx. hs s sme s, f [ A ] [ A], the s symmetrc mtrx. Exmple Gve exmple of symmetrc mtrx... 6..5 8 6 8 9. s symmetrc mtrx s., 6 d 8. Wht s skew-symmetrc mtrx? A mtrx s skew symmetrc f j for,..., d j,...,. hs s sme s A A [ ] [ ]. Exmple Gve exmple of skew-symmetrc mtrx. 0 0 5 5 0 s skew-symmetrc s ; ; 5. Sce oly f 0, ll the dgol elemets of skew-symmetrc mtrx hve to be zero. Wht s the trce of mtrx? he trce of mtrx s the sum of the dgol etres of, tht s, tr [ A] Exmple 4 Fd the trce of 5 6 4 7 6
Ury trx Opertos 04.04. tr [ A ] ( 5) + ( 4) + (6) 7 Exmple 5 he sles of tres re gve by mke (rows) d qurters (colums) for Blowout r us store locto A, s show below. 5 0 5 0 5 5 6 6 7 7 where the rows represet the sle of restoe, chg d Copper tres, d the colums represet the qurter umber,,, 4. Fd the totl yerly reveue of store A f the prces of tres vry by qurters s follows..5 0.0 5.0 0.05 [B] 40.9 8.0 4.0 8. 5.0.0 7.0.95 where the rows represet the cost of ech tre mde by restoe, chg d Copper, d the colums represet the qurter umbers. o fd the totl tre sles of store A for the whole yer, we eed to fd the sles of ech brd of tre for the whole yer d the dd to fd the totl sles. o do so, we eed to rewrte the prce mtrx so tht the qurters re rows d the brd mes re the colums, tht s, fd the trspose of [B]. [ C ] [ B].5 0.0 5.0 0.05 40.9 8.0 4.0 8. 5.0.0 7.0.95.5 40.9 5.0 0.0 8.0.0 [C] 5.0 4.0 7.0 0.05 8..95 Recogze ow tht f we fd [ A ][ C], we get D A C [ ] [ ][ ]
04.04.4 Chpter 04.04.5 40.9 5.0 5 0 0.0 8.0.0 5 0 5 5 5.0 4.0 7.0 6 6 7 7 0.05 8..95 597 965 9 74 5 5 76 69 he dgol elemets gve the sles of ech brd of tre for the whole yer, tht s d $597 (restoe sles) d $5 (chg sles) d $ (Cooper sles) he totl yerly sles of ll three brds of tres re d 597 + 5 + $5060 d ths s the trce of the mtrx. Defe the ermt of mtrx. he ermt of squre mtrx s sgle uque rel umber correspodg to mtrx. For mtrx, ermt s deoted by A or (A ). So do ot use d A terchgebly. For mtrx, [ A ] ( A) How does oe clculte the ermt of y squre mtrx? Let be mtrx. he mor of etry s deoted by d s defed s the ermt of the ( ( ) submtrx of, where the submtrx s obted by deletg the or th row d ( A) ( ) j th j colum of the mtrx. he ermt s the gve by + j for y,,, ( A) ( ) + j for y j,,,
Ury trx Opertos 04.04.5 coupled wth tht ( A ) for mtrx [ A], s we c lwys reduce the ermt + j of mtrx to ermts of mtrces. he umber ( ) s clled the cofctor of d s deoted by c. he bove equto for the ermt c the be wrtte s or ( A) C for y,,, j ( A) C for y j,,, he oly reso why ermts re ot geerlly clculted usg ths method s tht t becomes computtolly tesve. For mtrx, t requres rthmetc opertos proportol to!. Exmple 6 Fd the ermt of 5 5 64 8 44 ethod : ( A ) ( ) j Let the formul ( A ) ( ) j + j + j j j for y,, + + + ( ) + ( ) + ( ) + 5 5 64 8 44 8 4 5 5 64 8 44
04.04.6 Chpter 04.04 64 44 80 5 5 64 8 44 64 8 44 84 ( A ) + 5 4 5 80 + 84 00 + 400 84 84 Also for, ethod : ( A ) j + ( ) ( ) ( ) jc ( ) j C 4 + C ( ) 80 + C ( ) 84 ( A ) C + C + C ( 5) 4 + (5) 80 + () 84 00 + 400 84 84 ( ) ( ) + j ( ) ( ) ( ) A for y j,, Let j the formul ( A ) ( ) + + + + ( ) + ( ) + ( ) +
Ury trx Opertos 04.04.7 5 5 64 8 44 64 44 80 5 5 64 8 44 5 44 9 5 5 64 8 44 5 64 9 ( A) + 5( 80) + 8( 9) ( 9) 400 95 + 468 84 I terms of cofctors for j, ( A ) C + ( ) C 80 + C ( ) 9 + C ( ) 9 ( A ) C + C + C ( 5) 80 + (8) 9 + () 9 400 95 + 468 ( ) ( ) ( )
04.04.8 Chpter 04.04 84 Is there reltoshp betwee (AB), d (A) d (B)? Yes, f d [B] re squre mtrces of sme sze, the ( AB ) ( A)( B) Are there some other theorems tht re mportt fdg the ermt of squre mtrx? heorem : If row or colum mtrx s zero, the ( A ) 0. heorem : Let be mtrx. If row s proportol to other row, the ( A ) 0. heorem : Let be mtrx. If colum s proportol to other colum, the ( A ) 0. heorem 4: Let be mtrx. If colum or row s multpled by k to result mtrx k, the ( B ) k ( A). heorem 5: Let be upper or lower trgulr mtrx, the ( B) π. Exmple 7 Wht s the ermt of 0 6 0 7 4 0 4 9 5 0 5 Sce oe of the colums (frst colum the bove exmple) of s zero, ( A ) 0. Exmple 8 Wht s the ermt of 6 4 7 6 5 4 0 9 5 8 (A ) s zero becuse the fourth colum
Ury trx Opertos 04.04.9 4 6 0 8 s tmes the frst colum 5 9 Exmple 9 If the ermt of 5 5 64 8 44 s 84, the wht s the ermt of 5 0.5 [B] 64 6.8 44 5. Sce the secod colum of [B] s. tmes the secod colum of ( B ).( A) (.)( 84) 76.4 Exmple 0 Gve the ermt of 5 5 64 8 44 s 84, wht s the ermt of 5 5 [B] 0 4.8.56 44
04.04.0 Chpter 04.04 Sce [B] s smply obted by subtrctg the secod row of by.56 tmes the frst row of, (B) (A) 84 Exmple Wht s the ermt of 5 5 0 4.8.56 0 0 0.7 Sce s upper trgulr mtrx ( A ) ( )( )( ) ( 5)( 4.8)( 0.7) 84 Key erms: rspose Symmetrc trx Skew-Symmetrc trx rce of trx Determt