![Find matrix X given that AX - X = [(-4),(-10)] where A = [(2,3),(-2,4)]](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1786463510450-24eaf91671b210d1.png&w=3840&q=75)
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Hapa kuna hatua za kutatua swali la 5: a) Kokotoa thamani ya x inayowakilisha urefu wa kodi NK. Step 1: Tumia kanuni ya makutano ya nje ya kodi (Power of a Point Theorem). Kwa kodi mbili SM na NK zinazokutana nje kwenye P, bidhaa ya sehemu za kodi moja ni sawa na bidhaa ya sehemu za kodi nyingine. Hivyo, PM × PS = PK × PN. Step 2: Badilisha thamani ulizopewa: Tunajua SM = 15 cm, MP = 18 cm, PK = 22 cm, na NK = x cm. PS = PM + SM = 18 + 15 = 33 cm. PN = PK + NK = 22 + x cm. Badilisha hizi kwenye kanuni: 18 × 33 = 22 × (22 + x) Step 3: Tatua kwa x: 594 = 22(22 + x) Gawanya pande zote mbili kwa 22: (594)/(22) = 22 + x 27 = 22 + x Toa 22 kutoka pande zote mbili: x = 27 - 22 x = 5 cm Thamani ya x ni 5 cm. b) Kokotoa rediasi ya duara. Step 1: Kwa kuwa SOK ni kipenyo cha duara, pembe yoyote inayoundwa na kipenyo kwenye mzingo wa duara ni 90^. Hivyo, SNK = 90^ na SMK = 90^. Hii inamaanisha SNK na SMK ni pembetatu za pembe mraba. Step 2: Tumia Theorem ya Pythagoras kwenye SNK na SMK. Katika SNK: SN^2 + NK^2 = SK^2. Tunajua NK = 5 cm (kutoka sehemu a)) na SK = 2r (kipenyo ni mara mbili ya rediasi). SN^2 + 5^2 = (2r)^2 SN^2 + 25 = 4r^2 (Mlinganyo 1) Katika SMK: SM^2 + MK^2 = SK^2. Tunajua SM = 15 cm na SK = 2r. 15^2 + MK^2 = (2r)^2 225 + MK^2 = 4r^2 (Mlinganyo 2) Step 3: Tumia Kanuni ya Kosine (Law of Cosines) kwenye PSN na PMK ili kueleza SN^2 na MK^2 kwa kutumia ( P). Katika PSN: PS = 33 cm na PN = 22 + 5 = 27 cm. SN^2 = PS^2 + PN^2 - 2 · PS · PN · ( P) SN^2 = 33^2 + 27^2 - 2 · 33 · 27 · ( P) SN^2 = 1089 + 729 - 1782 ( P) SN^2 = 1818 - 1782 ( P) (Mlinganyo 3) Katika PMK: PM = 18 cm na PK = 22 cm. MK^2 = PM^2 + PK^2 - 2 · PM · PK · ( P) MK^2 = 18^2 + 22^2 - 2 · 18 · 22 · ( P) MK^2 = 324 + 484 - 792 ( P) MK^2 = 808 - 792 ( P) (Mlinganyo 4) Step 4: Linganisha Mlinganyo 1 na 3, na Mlinganyo 2 na 4. Kutoka Mlinganyo 1 na 3: 4r^2 - 25 = 1818 - 1782 ( P) 4r^2 = 1843 - 1782 ( P) (Mlinganyo 5) Kutoka Mlinganyo 2 na 4: 4r^2 - 225 = 808 - 792 ( P) 4r^2 = 1033 - 792 ( P) (Mlinganyo 6) Step 5: Tatua mfumo wa milinganyo 5 na 6. Linganisha Mlinganyo 5 na 6: 1843 - 1782 ( P) = 1033 - 792 ( P) Kusanya maneno ya ( P) upande mmoja na namba upande mwingine: 1843 - 1033 = 1782 ( P) - 792 ( P) 810 = 990 ( P) ( P) = (810)/(990) = (81)/(99) = (9)/(11) Sasa badilisha thamani ya ( P) kwenye Mlinganyo 6: 4r^2 = 1033 - 792 ((9)/(11)) 4r^2 = 1033 - (72 × 9) 4r^2 = 1033 - 648 4r^2 = 385 Step 6: Kokotoa r. r^2 = (385)/(4) r = sqrt((385)/(4)) r = sqrt(385)2 cm Rediasi ya duara ni sqrt(385)2 cm. Tuma swali linalofuata 📸