Lesson 55 - Inverse of Matrices & Determinants

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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 T-8/ 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 T-8/ Use the T-8/ 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 T-8 to investigte) s x - = - x =? (use T-8 to investigte)

3 // (G) Properties of nverses (n Mtrix Multiplition) re these properties true for (i) rel numbers? (ii) mtries? Use T-8 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 T-8/ to etermine the following prouts: Mth Honors - Sntowski () Determinnts n nvestigtion Use your T-8/ 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 T-8/ 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 T-8/ 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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