Lesson 55  Inverse of Matrices & Determinants


 Clinton Welch
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1 // () Review Lesson  nverse of Mtries & Determinnts Mth Honors  Sntowski  t this stge of stuying mtries, we know how to, subtrt n multiply mtries i.e. if Then evlute: () + B (b)  () B () B (e) B n B Mth Honors  Sntowski Mth Honors  Sntowski (B) Review of Rel Numbers (C) Strtegy for Diviing Mtries if we ivie by 8 (i.e. /8), we oul rerrnge n look t ivision s nothing more thn simple multiplition thus /8 = x /8 = x 8  so in wy, we woul never hve to perform ivision s long s we simply multiply by the inverse (or reiprol) One other note bout this inverse of number number n its inverse (its reiprol) hve the property tht (n) x (n  ) =  i.e. (8) (8  ) = (8) (/8) = (8/8) = So how oes multiplitive inverses relte to DVSON of MTRCES f number n its inverse (its reiprol) hve the property tht (n) x (n  ) = Then. So how oes this relte to DVSON of MTRCES Mth Honors  Sntowski Mth Honors  Sntowski (C) Strtegy for Diviing Mtries (C) Strtegy for Diviing Mtries So how oes multiplitive inverses relte to DVSON of MTRCES f number n its inverse (its reiprol) hve the property tht (n) x (n  ) = Then. mtrix n its inverse shoul hve the property tht B x B  = So. mtrix n its inverse shoul hve the property tht B x B  = Well wht is in terms of mtries? simply the ientity mtrix, Thus B x B  = or Mth Honors  Sntowski Mth Honors  Sntowski 6
2 // Mth Honors  Sntowski 7 (D) nverse Mtries Given mtrix, whih of the following is the inverse of mtrix? E D C B Mth Honors  Sntowski 8 (D) nverse Mtries Solve for x: then, 7 Let x Mth Honors  Sntowski 9 (E) Terms ssoite with nverse Mtries Thus we hve new terms tht relte to inverse mtries: () mtrix is invertible if it hs n inverse (b) mtrix is singulr if it oes NOT hve n inverse Mth Honors  Sntowski (F) nverse Mtries on T8/ So we hve the bsi ie of inverse mtries how n use the lultor to fin the inverse of mtrix Mth Honors  Sntowski (F) nverse Mtries on T8/ Use the T8/ to etermine the inverse of: B Mth Honors  Sntowski (G) Properties of nverses (n Mtrix Multiplition) s multiplition with rel numbers ommuttive (is b = b)? s mtrix multiplition ommuttive s B = B? (use T8 to investigte) s x  =  x =? (use T8 to investigte)
3 // (G) Properties of nverses (n Mtrix Multiplition) re these properties true for (i) rel numbers? (ii) mtries? Use T8 to investigte s (  )  =? s (B)  =  B ? (H) Determining the nverse of Mtrix How n we etermine the inverse of mtrix if we DO NOT hve ess to our lultors? (i) Mtrix Multiplition (ii) Clulting the eterminnt Mth Honors  Sntowski Mth Honors  Sntowski (H) Determining the nverse of Mtrix (H) Determining the nverse of Mtrix Let s use Mtrix Multiplition to fin the inverse of So our mtrix will be b n we now hve the multiplition b n so using our knowlege of mtrix multiplition, we get n so using our knowlege of mtrix multiplition, we get system of equtions b b so Whih we n solve s: so so n b n n b so n b b Mth Honors  Sntowski Mth Honors  Sntowski 6 (H) Determining the nverse of Mtrix (H) Determining the nverse of Mtrix So if How n we etermine the inverse of mtrix if we DO NOT hve ess to our lultors? So our mtrix will be (ii) Clulting the eterminnt So Metho # involve something lle eterminnt whih mens.. Mth Honors  Sntowski 7 Mth Honors  Sntowski 8
4 // Mth Honors  Sntowski 9 () Determinnts n nvestigtion Use your T8/ to etermine the following prouts: Mth Honors  Sntowski () Determinnts n nvestigtion Use your T8/ to etermine the following prouts: Mth Honors  Sntowski () Determinnts n nvestigtion Now refully look t the mtries you multiplie n observe pttern Mth Honors  Sntowski () Determinnts n nvestigtion Now refully look t the mtries you multiplie n observe pttern b? Mth Honors  Sntowski () Determinnts n nvestigtion Now PROVE your pttern hols true for ll vlues of, b,,. Mth Honors  Sntowski () Determinnts n nvestigtion Now PROVE your pttern hols true for ll vlues of, b,,. b b b b
5 // () Determinnts n nvestigtion () Determinnts n nvestigtion So to summrize: b b OR b b then we see tht from our originl mtrix, the vlue () hs speil signifine, in tht its vlue etermines whether or not mtrix n be inverte if  oes not equl, mtrix woul be lle "invertible  i.e. if  =, then mtrix nnot be inverte n we ll it singulr mtrix  the vlue  hs speil nme it will be lle the eterminnt of mtrix n hs the nottion et or Mth Honors  Sntowski Mth Honors  Sntowski 6 () Determinnts n nvestigtion (J) Exmples So if is invertible then b then b where ex. Fin the eterminnt of the following mtries n hene fin their inverses: Verify using T8/ B Mth Honors  Sntowski 7 Mth Honors  Sntowski 8 (J) Exmples (L) Homework ex. Fin the eterminnt of the following mtries n hene fin their inverses: Verify using T8/ C 7 8 B D 6 HW S.; p9; Q,6,8,9,o, Mth Honors  Sntowski 9 Mth Honors  Sntowski
6 // (J) Exmples Prove whether the following sttements re true or flse for by mtries. Remember tht ounterexmple estblishes tht sttement is flse. n generl, you my NOT ssume tht sttement is true for ll mtries beuse it is true for by mtries, but for the exmples in this question, those tht re true for by mtries re true for ll mtries if the imensions llow the opertions to be performe. Questions: () et B etetb (b) et () et () et et B et etb T et Mth Honors  Sntowski 6
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