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 hesabu za vigezo (determinants) kwa kutumia upanuzi wa cofactor: Step 1: Kokotoa kigezo cha matrix A Matrix A imetolewa kama: A = 4 & 3 & 0 \\ 6 & 5 & 2 \\ 9 & 7 & 3 Tutatumia upanuzi wa cofactor kwenye safu ya kwanza: (A) &= 4 · 5 & 2 \\ 7 & 3 - 3 · 6 & 2 \\ 9 & 3 + 0 · 6 & 5 \\ 9 & 7 \\ &= 4((5)(3) - (2)(7)) - 3((6)(3) - (2)(9)) + 0 \\ &= 4(15 - 14) - 3(18 - 18) + 0 \\ &= 4(1) - 3(0) + 0 \\ &= 4 Kigezo cha A ni 4. Step 2: Kokotoa kigezo cha matrix B Matrix B imetolewa kama: B = 6 & 0 & 0 & 5 \\ 1 & 7 & 2 & -5 \\ 2 & 0 & 0 & 0 \\ 8 & 3 & 1 & 8 Tutatumia upanuzi wa cofactor kwenye safu ya tatu kwa sababu ina namba nyingi za sifuri: (B) &= 2 · (-1)^3+1 0 & 0 & 5 \\ 7 & 2 & -5 \\ 3 & 1 & 8 + 0 + 0 + 0 \\ &= 2 · 0 & 0 & 5 \\ 7 & 2 & -5 \\ 3 & 1 & 8 Sasa, kokotoa kigezo cha matrix ndogo ya 3x3 kwa kutumia upanuzi wa cofactor kwenye safu ya kwanza: 0 & 0 & 5 \\ 7 & 2 & -5 \\ 3 & 1 & 8 &= 0 - 0 + 5 · (-1)^1+3 7 & 2 \\ 3 & 1 \\ &= 5((7)(1) - (2)(3)) \\ &= 5(7 - 6) \\ &= 5(1) \\ &= 5 Kwa hiyo, kigezo cha B ni: (B) = 2 · 5 = 10 Kigezo cha B ni 10. Step 3: Kokotoa kigezo cha matrix C Matrix C imetolewa kama: C = 4 & 0 & 0 & 0 & 0 \\ 7 & -1 & 0 & 0 & 0 \\ 2 & 6 & 3 & 0 & 0 \\ 5 & -8 & -4 & -3 & 0 \\ 5 & 0 & 0 & 0 & 2 Tutatumia upanuzi wa cofactor kwenye safu ya kwanza: (C) &= 4 · (-1)^1+1 -1 & 0 & 0 & 0 \\ 6 & 3 & 0 & 0 \\ -8 & -4 & -3 & 0 \\ 0 & 0 & 0 & 2 + 0 + 0 + 0 + 0 \\ &= 4 · -1 & 0 & 0 & 0 \\ 6 & 3 & 0 & 0 \\ -8 & -4 & -3 & 0 \\ 0 & 0 & 0 & 2 Sasa, kokotoa kigezo cha matrix ndogo ya 4x4 kwa kutumia upanuzi wa cofactor kwenye safu ya kwanza: -1 & 0 & 0 & 0 \\ 6 & 3 & 0 & 0 \\ -8 & -4 & -3 & 0 \\ 0 & 0 & 0 & 2 &= -1 · (-1)^1+1 3 & 0 & 0 \\ -4 & -3 & 0 \\ 0 & 0 & 2 + 0 + 0 + 0 \\ &= -1 · 3 & 0 & 0 \\ -4 & -3 & 0 \\ 0 & 0 & 2 Kisha, kokotoa kigezo cha matrix ndogo ya 3x3 kwa kutumia upanuzi wa cofactor kwenye safu ya kwanza: 3 & 0 & 0 \\ -4 & -3 & 0 \\ 0 & 0 & 2 &= 3 · (-1)^1+1 -3 & 0 \\ 0 & 2 + 0 + 0 \\ &= 3((-3)(2) - (0)(0)) \\ &= 3(-6 - 0) \\ &= 3(-6) \\ &= -18 Kwa hiyo, kigezo cha C ni: (C) = 4 · (-1) · (-18) = 4 · 18 = 72 Kigezo cha C ni 72. Tuma swali linalofuata 📸