Diketahui matriks A=(2 -5 -5 12) dan B=(1 -2 -1 1).

a. (2A+B)^(-1) = [[-25/7, -12/7], [-11/7, -5/7]], b. A^(-1) (B+3A)^(-1) = [[524/13, 239/13], [217/13, 99/13]]

Pembahasan

Untuk menyelesaikan soal ini, kita perlu menghitung dua bagian terpisah:

a. (2A + B)^(-1)
b. A^(-1) (B + 3A)^(-1)

Diketahui:
Matriks A = [[2, -5], [-5, 12]]
Matriks B = [[1, -2], [-1, 1]]

Langkah 1: Hitung 2A
2A = 2 * [[2, -5], [-5, 12]] = [[4, -10], [-10, 24]]

Langkah 2: Hitung 2A + B
2A + B = [[4, -10], [-10, 24]] + [[1, -2], [-1, 1]] = [[4+1, -10-2], [-10-1, 24+1]] = [[5, -12], [-11, 25]]

Langkah 3: Hitung invers dari (2A + B), yaitu (2A + B)^(-1)
Untuk matriks [[a, b], [c, d]], inversnya adalah (1/(ad-bc)) * [[d, -b], [-c, a]].
Determinan dari (2A + B) = (5 * 25) - (-12 * -11) = 125 - 132 = -7.
(2A + B)^(-1) = (1/-7) * [[25, 12], [11, 5]] = [[-25/7, -12/7], [-11/7, -5/7]]

Langkah 4: Hitung A^(-1)
Determinan dari A = (2 * 12) - (-5 * -5) = 24 - 25 = -1.
A^(-1) = (1/-1) * [[12, 5], [5, 2]] = [[-12, -5], [-5, -2]]

Langkah 5: Hitung 3A
3A = 3 * [[2, -5], [-5, 12]] = [[6, -15], [-15, 36]]

Langkah 6: Hitung B + 3A
B + 3A = [[1, -2], [-1, 1]] + [[6, -15], [-15, 36]] = [[1+6, -2-15], [-1-15, 1+36]] = [[7, -17], [-16, 37]]

Langkah 7: Hitung invers dari (B + 3A), yaitu (B + 3A)^(-1)
Determinan dari (B + 3A) = (7 * 37) - (-17 * -16) = 259 - 272 = -13.
(B + 3A)^(-1) = (1/-13) * [[37, 17], [16, 7]] = [[-37/13, -17/13], [-16/13, -7/13]]

Langkah 8: Hitung A^(-1) * (B + 3A)^(-1)
A^(-1) * (B + 3A)^(-1) = [[-12, -5], [-5, -2]] * [[-37/13, -17/13], [-16/13, -7/13]]
= [[(-12*-37/13)+(-5*-16/13), (-12*-17/13)+(-5*-7/13)], [(-5*-37/13)+(-2*-16/13), (-5*-17/13)+(-2*-7/13)]]
= [[(444/13)+(80/13), (204/13)+(35/13)], [(185/13)+(32/13), (85/13)+(14/13)]]
= [[524/13, 239/13], [217/13, 99/13]]

Jawaban:
a. (2A+B)^(-1) = [[-25/7, -12/7], [-11/7, -5/7]]
b. A^(-1) (B+3A)^(-1) = [[524/13, 239/13], [217/13, 99/13]]