Chapter Four: Matrices Theory
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1 Uiversity of echology Egieerig lysis ecture otes Dep. Of Electricl & Electroic Eg. hird yer ecture ec. Dr. bbs H. Iss hpter Four: trices heory Refereces:. dvced Egieerig themtics by. Ry Wylie. dvced Egieerig themtics by Erwi Kreyszig. Defiitio: mtrix of order m x, or m by is rectgulr mtrix, rry of umbers hvig m rows d colums. It c be writte i the form m m m R If m it is clled squre mtrix of order m or.. Digol mtrix: squre mtrix D is sid to be digol mtrix if the elemet of the mtrix stisfy D j i d j i d mtrix is clled lower trigulr mtrix of order mtrix U is clled upper trigulr mtrix of order U
2 Uiversity of echology Egieerig lysis ecture otes Dep. Of Electricl & Electroic Eg. hird yer ecture ec. Dr. bbs H. Iss. Equlity of mtrices: wo mtrices d B b of the sme order re equl iff. b. dditio d subtrctio of mtrices: o dd or subtrct, two mtrices must be of the sme order if [ ] B [ b ] B [ m b ], m the Note tht +B B+ ommuttive lw +B + + B+ ssocitive lw. ultiplictio of mtrix by sclr: If [ ] d q is sclr umber, the q q [ ]. ultiplictio of mtrices: let m r p [ ] R d B [ b ] R he B is defied oly whe r d it is m x p mtrix [ c ], c i >>> Properties of mtrix opertio: k B k B ssocitive B B B d + B + B distributive + B + B B B Not commultive ore over B Not ecessry imply or B ji b ik. B B is pre-multiplied by or is post- multiplied by B
3 Uiversity of echology Egieerig lysis ecture otes Dep. Of Electricl & Electroic Eg. hird yer ecture ec. Dr. bbs H. Iss. rspose of mtrix: If the Note tht kj + B + B B B d.8 Symmetric d skew symmetric mtrices: If symmetric mtrix is squre mtrix If - skew- symmetric mtrix e.g. B is symmetric.9 Priciple digol & trce: is skew symmetric If [ ] is squre mtrix, the the digol which cotis ll elemets of is clled the priciple or mi digol. he sum of these elemets is clled the trce e.g.. Uity mtrix:, hetrce of + + It is squre mtrix tht ll elemets of its priciple digol re while the other is I I. Determit: If is squre mtrix the is the determit of. foud s follows: - ior give y elemet of of we ssocite ew determit of order - obtied by removig ll elemets of the J row d K colum. his is clled the ior of
4 Uiversity of echology Egieerig lysis ecture otes Dep. Of Electricl & Electroic Eg. hird yer ecture ec. Dr. bbs H. Iss Exmple: he mior of elemet is i b- ofctor: if we multiply the mior of by - j+k the result is clled the cofctor of d it is deoted by e.g. the cofctor of elemet i lst exmple is + Now Exmple: det Δ k JK is order of Δ djoit of mtrix: If is x squre mtrix defied s We c form ew mtrix of the cofctors Where is the cofctors of or is the cofctor of
5 Uiversity of echology Egieerig lysis ecture otes Dep. Of Electricl & Electroic Eg. hird yer ecture ec. Dr. bbs H. Iss d if we tke the trspose of the is clled the djoit of the origil mtrix dj. Exmple: + +, the d dj 8. trix Iversio: If is o-sigulr mtrix of order i.e, the there exists uique iverse - such tht - d c be expressed s Exmple: from previous exmple det Note tht:. he iverse of x mtrix 8 is det. he iverse of o-sigulr digol mtrix is
6 Uiversity of echology Egieerig lysis ecture otes Dep. Of Electricl & Electroic Eg. hird yer ecture ec. Dr. bbs H. Iss / / / Exmple: / /.. P Q PQ or. Rk: he rk of mtrix is the lrgest vlue of r for which there exists r x r sub mtrix of with o-vishig determit. Exmple: he mtrix, is of the rk, sice ech of the third-order sub mtrices,,,, is sigulr, while ot ll secod-order sub mtrices re sigulr.. Elemetry opertio: Gussi elimitio method. Iterchgig colums or rows. ultiplictio of row colum by o-zero umber.. dditio to or subtrctio from ll the elemets of y row colum k times the correspodig elemets of y other row colum. We will use elemetry row opertio oly ERO:
7 Uiversity of echology Egieerig lysis ecture otes Dep. Of Electricl & Electroic Eg. hird yer ecture ec. Dr. bbs H. Iss <<<>>> Iverse of mtrix by usig ERO, Exmple: fid the iverse of Solutio: write dow the ugmeted mtrix R R R -R R -R - R -R R +R x-/ R -/ R R -/ R / / / / / / / / 8 / / / / / / / / / 8 / / / / / / / / / / / / / / / / / / / / / /
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