Linear Algebra. Definition The inverse of an n by n matrix A is an n by n matrix B where, Properties of Matrix Inverse. Minors and cofactors

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1 Dfnton Th nvr of an n by n atrx A an n by n atrx B whr, Not: nar Algbra Matrx Invron atrc on t hav an nvr. If a atrx ha an nvr, thn t call. Proprt of Matrx Invr. If A an nvrtbl atrx thn t nvr unqu.. (A - ) -. (A k ) -. (ca) -. ( A T ) - 6. If A an nvrtbl atrx, thn th yt of quaton Ax b ha a unqu oluton gvn by Th nvr of a x atrx A x atrx a A c b A x atrx A f an only f a-bc. Th quantty (a-bc) ha o othr uful proprt a wll an o fn to b th of th atrx A. Mnor an cofactor If A a quar atrx, thn th M j of th lnt a j of A th trnant of th atrx obtan by ltng th -th row an th j-th colun fro A. Th j (-) j M j.

2 Dfnton of a Dtrnant If A a quar atrx of orr or gratr, thn th trnant of A th u of th ntr n th frt row* of A ultpl by thr cofactor. That, * any can b u a th pvot for th cofactor o not hav to b frt row Dtrnant of -by- Matrx A j a a A c b j j a a ( ) M t( A) A a b ( c) ( ) M Dtrnant of a x atrx Dtrnant of -by- Matrx If A a x atrx Thn w fn a a a t(a) a a a a a a a A a a a a a a a a a a a a a -a a a a a a a a a a A t( A ) A a h a g f b c f h b g pvot row f c g h

3 Dtrnant of -by- Matrx Dtrnant of a x atrx A t( A ) A a h a g f b c f h b h pvot colun c b g c f Exapl - t(a) Exapl: valuat t(a) for: A t(a)() - -() (-) - ()()-()(6)-(-)()98 In-la Exrc # Fn t(a) ung a) cofactor xpanon along th frt row. b) cofactor xpanon along th frt colun. c) cofactor xpanon along th con colun.

4 Proprt of Dtrnant. Th valu of a trnant f t row ar wrttn a colun n th a orr If any two row (or two colun) of a trnant ar ntrchang, th valu of th trnant Proprt of Dtrnant. A coon factor of all lnt of any row (or colun) th trnant. 8 Proprt of Dtrnant. If th corrponng lnt of two row (or colun) of a trnant ar, th valu of th trnant Matrx Invron Ajont Mtho How to calculat th atrx nvr? Manng: Row ( Row ) on Row ( Row ). Thrfor, th lnar yt wth thr unknown o not hav a unqu oluton.

5 Mnor an ofactor of a Matrx Entry Mnor an ofactor of a Matrx Entry For x atrx a a a A a a a a a a For x atrx a a a A a a a a a a M (-) M M (-) M for t row, t colun for t row, t colun for n row, r colun for n row, r colun Ajont Matrx - ofactor Ajont Matrx Mnor an ofactor Th ajont atrx of [A], aj[a] obtan by takng th tranpo of th cofactor atrx of [A]. A aj ( A) A [ ] Matrx Invron onr th followng t of ultanou lnar quaton. Th coffcnt can b arrang n a atrx for a hown.

6 Matrx Invron Matrx Invron Th rultng atrx of nor : Matrx Invron ofactor: ofactor ar th gn nor. Th cofactor of lnt a j of atrx [A] : Thrfor Th rultng cofactor atrx of : Matrx Invron Ajont atrx: Th ajont atrx of [A], aj[a] obtan by takng th tranpo of th cofactor atrx of [A].

7 Matrx Invron In-la Exrc # U th ajont tho to fn th nvr of th A atrx: 6 z y x ME 7 - Tranfr Functon Exapl (t) - R Fn th tranfr functon for th lctrcal yt blow t c ( ) R - - (t) t c Soluton tchnqu to tak th aplac Tranfor of th quaton, t ntal conton to zro, an olv th SE ME 7 - Tranfr Functon Exapl [ ] ) ( E I E R Sultanou lnar quaton wth ybolc coffcnt, Wll n to fn th trnant of th x atrx, u a procur a for nurcal coffcnt.

8 ME 7 - Tranfr Functon Exapl R In-la Exrc # U th ajont tho to fn th nvr of th followng atrx: Δ B K K τ ω ω θ ( )( ) ( ) ( ) K B K B M Matrx Invron Ung G- Elnaton If Gau oran lnaton appl on a quar atrx, t can b u to calculat th atrx nvr. Th can b on by augntng th quar atrx wth th ntty atrx of th a non, an through th followng atrx opraton: Exapl A If th orgnal quar atrx, A, gvn by th followng xpron: Thn, aftr augntng by I, th followng obtan:

9 Exapl By prforng lntary row opraton on th [AI] atrx untl A rach ruc row chlon for, th followng th fnal rult: [ I A ] Th atrx augntaton can now b unon, whch gv th followng: I Atonal Rourc A Txtbook haptr on Syt of Equaton Duplcat olun Mtho for x atrx Fro th prvou l: a a a t(a) a a a a a a a a a a a -a a a a a a a a a a t(a) a a a a a a a a a -a a a -a a a -a a a Duplcat olun Mtho for x Exapl : - t(a) - a a a a a a a a a Alo work wth x, x, tc.