![Given that A = [8/(x+1), 2] and B = [1, 2] are singular matrices, find the value of x.](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1782211188453-dfe2313b24979049.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.
ScanSolve AI Answer
Haya, nitasuluhisha swali hili. Tunapewa matrisi A = 8 & x \\ x+1 & 2 na B = 1 & 2 \\ 3 & 0 . Tunahitaji kupata thamani ya x ambayo AB ni matrisi isiyo na kigeuzi (singular matrix). Matrisi isiyo na kigeuzi ina kigeuzi (determinant) cha sifuri. Step 1: Tafuta matrisi ya AB. Ili kuzidisha matrisi, tunazidisha safu mlalo za matrisi ya kwanza na safu wima za matrisi ya pili. AB = 8 & x \\ x+1 & 2 1 & 2 \\ 3 & 0 AB = (8)(1) + (x)(3) & (8)(2) + (x)(0) \\ (x+1)(1) + (2)(3) & (x+1)(2) + (2)(0) AB = 8 + 3x & 16 + 0 \\ x+1 + 6 & 2x+2 + 0 AB = 3x+8 & 16 \\ x+7 & 2x+2 Step 2: Tafuta kigeuzi (determinant) cha AB. Kwa matrisi ya 2 × 2, a & b \\ c & d , kigeuzi ni ad - bc. Hivyo, kigeuzi cha AB ni: (AB) = (3x+8)(2x+2) - (16)(x+7) Panua maneno: (AB) = (6x^2 + 6x + 16x + 16) - (16x + 112) (AB) = 6x^2 + 22x + 16 - 16x - 112 Unganisha maneno yanayofanana: (AB) = 6x^2 + (22x - 16x) + (16 - 112) (AB) = 6x^2 + 6x - 96 Step 3: Weka kigeuzi sawa na sifuri na utatue kwa x. Kwa kuwa AB ni matrisi isiyo na kigeuzi, (AB) = 0. 6x^2 + 6x - 96 = 0 Gawanya mlinganyo mzima kwa 6 ili kurahisisha: (6x^2)/(6) + (6x)/(6) - (96)/(6) = 0 x^2 + x - 16 = 0 Huu ni mlinganyo wa quadratic. Tutatumia fomula ya quadratic x = -b ± sqrt(b^2 - 4ac)2a. Hapa, a=1, b=1, c=-16. x = -1 ± sqrt(1^2 - 4(1)(-16))2(1) x = -1 ± sqrt(1 + 64)2 x = -1 ± sqrt(65)2 Kwa hivyo, kuna thamani mbili za x: x_1 = -1 + sqrt(65)2 x_2 = -1 - sqrt(65)2 Thamani za x ni -1 + sqrt(65)2 na -1 - sqrt(65)2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.