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 suluhisho la swali lako: 20. a) Matrix A ni singular. Tafuta x na ueleze matrix A. Matrix A imetolewa kama: A = x & x-1 \\ 2 & 4 Matrix ni singular ikiwa determinant yake ni sifuri. Determinant ya matrix 2 × 2, a & b \\ c & d , ni ad - bc. Step 1: Kokotoa determinant ya matrix A. (A) = (x)(4) - (x-1)(2) (A) = 4x - (2x - 2) (A) = 4x - 2x + 2 (A) = 2x + 2 Step 2: Weka determinant sawa na sifuri na utatue kwa x. 2x + 2 = 0 2x = -2 x = (-2)/(2) x = -1 Step 3: Eleza matrix A kwa kutumia thamani ya x uliyopata. Badilisha x = -1 kwenye matrix A: A = -1 & (-1)-1 \\ 2 & 4 A = -1 & -2 \\ 2 & 4 b) Wafanyabiashara wawili, Juma na Wanjiku walinunua bidhaa kutoka kwa muuzaji wa jumla. i) Idadi ya bidhaa zilizonunuliwa na Juma na Wanjiku kwa kutumia matrix ya 2 × 3. Juma alinunua mikate 18, pakiti 40 za maziwa, na sabuni 5. Wanjiku alinunua mikate 15, pakiti 30 za maziwa, na sabuni 6. Matrix ya 2 × 3 (safu 2, nguzo 3) itawakilisha wanunuzi (Juma, Wanjiku) kama safu na bidhaa (mikate, maziwa, sabuni) kama nguzo. Idadi ya bidhaa = 18 & 40 & 5 \\ 15 & 30 & 6 ii) Bei za bidhaa zilizonunuliwa kwa kutumia matrix ya 3 × 1. Bei ya mkate ni Ksh. 45. Bei ya pakiti ya maziwa ni Ksh. 50. Bei ya sabuni ni Ksh. 150. Matrix ya 3 × 1 (safu 3, nguzo 1) itawakilisha bei za bidhaa. Bei za bidhaa = 45 \\ 50 \\ 150 iii) Jumla ya gharama iliyolipwa na kila mfanyabiashara. Ili kupata jumla ya gharama iliyolipwa na kila mfanyabiashara, tunazidisha matrix ya idadi ya bidhaa na matrix ya bei. Step 1: Zidisha matrix ya idadi ya bidhaa na matrix ya bei. Gharama = 18 & 40 & 5 \\ 15 & 30 & 6 45 \\ 50 \\ 150 Step 2: Kokotoa bidhaa ya matrix. Kwa Juma (safu ya kwanza): (18 × 45) + (40 × 50) + (5 × 150) = 810 + 2000 + 750 = 3560 Kwa Wanjiku (safu ya pili): (15 × 45) + (30 × 50) + (6 × 150) = 675 + 1500 + 900 = 3075 Step 3: Andika matokeo kama matrix. Gharama = 3560 \\ 3075 Hivyo, Juma alilipa Ksh. 3560 na Wanjiku alilipa Ksh. 3075. That's 4 down, 1 left today. What's next?