Mtri lger Mtri nverses Setion.. Mtri nverses hree si opertions on mtries, ition, multiplition, n sutrtion, re nlogues for mtries of the sme opertions for numers. n this setion we introue the mtri nlogue of numeril ivision. o egin, onsier hoe numeril eqution: is solve when n re known numers. f, there is no solution ( unless ). ut if, we n multipl oth sies inverse of, enote, to otin the solution. his multiplition is ommonl lle iviing, n the propert of work is tht tht mkes the. More over, we sw in the previous setion tht the role tht pls in rithmeti is ple in mtri lger the ientit mtri. his suggests the following efinition. f is squre mtri, mtri is lle n inverse of if n onl if: n mtri tht hs n inverse is lle n invertile mtri. Emple Show tht is n inverse of Solution: Compute n Hene,, so mtri is n inverse of Liner lger
Mtri lger Mtri nverses Emple Show tht hs no solution Solution: Let enote n ritrr mtri. hen: so hs row of zeros. Hene nnot equl for n. Emple shows tht it is possile for nonzero mtri to hve no inverse. ut if mtri hs n inverse, it hs onl one. heorem f mtri hs n inverse, it is unique Proof Let is mtri tht hs two inverse, these lle n C. Sine n C re oth inverses of, we hve C. Hene ( C) C( ) C C. f somehow mtri n e foun suh tht e,then is n invertile n is the inverse of, in smols,. Emple 6 f n, show tht Solution: We verif tht n we leve to the reer Liner lger
Mtri lger Mtri nverses Eerises:. Fin the inverse of the mtries s follow. ( Use the efinition of n inverse of mtri ):.. 6 8. Fin the inverse of the mtries s follow. ( Use the efinition of n inverse of mtri ).. nverse n Liner Sstem Mtri inverses n e use t o solve ertin sstems of liner equtions. sstem of liner equtions n e written s single mtri eqution: X where n re known mtries n X is to e etermine. f is invertile, then multipl eh sie of the eqution the left X X X to get: Liner lger
Mtri lger Mtri nverses his gives solution to the sstem of equtions, euse if X is sustitute to this eqution will e hol: X Furthermore, the rgument shows tht if X is n solution, then neessril X, so the solution is unique. Of ourse the tehnique works onl when the oeffiient mtri hs n inverse. heorem 8 Suppose sstem of n equtions in n vriles is written in mtri form s: f the n solution: X n oeffiient mtri is invertile, the sstem hs the unique X Emple Solve this sstem of liner equtions Solution: e written in mtri form: We get the inverse of the oeffiient mtri s follow: So the solution of sstem of liner equtions is: 6 Liner lger
Mtri lger Mtri nverses n nversion Metho eqution Given the invertile mtri, we etermine. Write from the where,,, re to e etermine. Equting olumns in the eqution gives n hese re sstems of liner equtions, eh with s oeffiient mtri. Sine is invertile, eh sstem hs unique solution heorem. ut this mens tht the reue row ehelon form R of nnot hve row of zeros, n so is the ientit mtri ( R is squre ). Hene, there is sequene of elementr row opertions rring to the ientit mtri. his sequene rries the ugmente mtries of oth sstem to reue row ehelon form n so solves the sstems: Hene, we n o oth lultions simultneousl. his n e written more omptl s follows: n other wors, the sequene of row opertions tht rries to lso rries to. Mtri nversion lgorithm f is n invertile mtri, there eists sequene of elementr row opertions tht rr to the ientit mtri of the sme size, written Liner lger 6
Mtri lger Mtri nverses Liner lger. his sme series of row opertions rries to, tht is. he lgorithm n e summrize s follows: where the row opertions on n re rrie out simultneousl. Emple 8 Use the inversion lgorithm to fin the inverse of the mtri: Solution ppl elementr row opertions to the oule mtri: So s to rr to. First interhnge rows n. Sutrt times row from row, n sutrt row from row Continue to reue row-ehelon form.
Mtri lger Mtri nverses Hene, heorem 9 f is n n n mtri, either n e reue to elementr row opertions or it nnot. n the first se, the lgorithm proues seon se, oes not eist. ; in the Properties of nverses heorem ll the following mtries re squre mtries of the sme size:. is invertile n. f is invertile, so is ), n (. f, re invertile, so is n ( ). f,,..., k re ll invertile mtries, so is the prout... k n (... k ) k.... f is invertile, so is k for k n k k ( ) ( ) 6. f is invertile n is numer, then is invertile n ( ). f is invertile, so is its trnspose n ( ) ( ) Corollr squre mtri is invertile if n onl if is invertile. Emple 9 Fin if ( ) Solution heorem () we otin: Liner lger 8
Mtri lger Mtri nverses ( ) [ ) ( ] - Hene +, so heorem he following onitions re equivlent for n n n mtri :. is invertile. he homogeneous sstem X hs onl the trivil solution X. n e rrie to the ientit mtri n elementr row opertions.. he sstem X hs t lest one solution X for ever hoie of olumn.. here eists n n n mtri C suh tht C n Corollr f n C re squre mtries suh tht C. n prtiulr, oth n C re invertile, C n C. Eerises.. Fin the inverse of eh of the following mtries:... 6. n eh se, solve the sstem of equtions fining the inverse of the oeffiient mtri: Liner lger 9
Mtri lger Mtri nverses. z z z. z z z. w z w w. Given. Solve the sstem of eqution X. Fin mtri suh tht. Fin when :. ( ).. ( ). f,, C re squre mtries n C, show tht is invertile n C 6. Verif tht stisfies n use this ft to show tht ( ). Let, enote n n invertile mtries:. ( ). f is invertile, so tht ( ) is invertile n fin formul for Liner lger