Diketahui matriks A=(2 1 0 3 -1 4 2 2 0) dan C=(8 7 45 23

Invers matriks A adalah [[1, 0, -1/2], [-1, 0, 1], [-1, 1/4, 5/8]]. Matriks B adalah [[3, -3, 6], [2, 13, 33], [4, -2, 6]].

Pembahasan

Untuk menyelesaikan soal ini, kita perlu melakukan dua langkah: mencari invers matriks A dan kemudian mencari matriks B.

Diketahui:
A = [[2, 1, 0], [3, -1, 4], [2, 2, 0]]
C = [[8, 7, 45], [23, -30, 9], [10, 20, 78]]

a. Mencari invers matriks A (A⁻¹):

Langkah 1: Hitung determinan matriks A (det(A)).
det(A) = 2 * det([[ -1, 4 ], [ 2, 0 ]]) - 1 * det([[ 3, 4 ], [ 2, 0 ]]) + 0 * det([[ 3, -1 ], [ 2, 2 ]])
det(A) = 2 * ((-1 * 0) - (4 * 2)) - 1 * ((3 * 0) - (4 * 2)) + 0
det(A) = 2 * (0 - 8) - 1 * (0 - 8)
det(A) = 2 * (-8) - 1 * (-8)
det(A) = -16 + 8
det(A) = -8

Karena determinannya tidak nol (det(A) = -8), maka invers matriks A ada.

Langkah 2: Hitung matriks kofaktor.
Kofaktor A₁₁ = (-1)^(1+1) * det([[-1, 4], [2, 0]]) = 1 * (0 - 8) = -8
Kofaktor A₁₂ = (-1)^(1+2) * det([[3, 4], [2, 0]]) = -1 * (0 - 8) = 8
Kofaktor A₁₃ = (-1)^(1+3) * det([[3, -1], [2, 2]]) = 1 * (6 - (-2)) = 8
Kofaktor A₂₁ = (-1)^(2+1) * det([[1, 0], [2, 0]]) = -1 * (0 - 0) = 0
Kofaktor A₂₂ = (-1)^(2+2) * det([[2, 0], [2, 0]]) = 1 * (0 - 0) = 0
Kofaktor A₂₃ = (-1)^(2+3) * det([[2, 1], [2, 2]]) = -1 * (4 - 2) = -2
Kofaktor A₃₁ = (-1)^(3+1) * det([[1, 0], [-1, 4]]) = 1 * (4 - 0) = 4
Kofaktor A₃₂ = (-1)^(3+2) * det([[2, 0], [3, 4]]) = -1 * (8 - 0) = -8
Kofaktor A₃₃ = (-1)^(3+3) * det([[2, 1], [3, -1]]) = 1 * (-2 - 3) = -5

Matriks kofaktor = [[-8, 8, 8], [0, 0, -2], [4, -8, -5]]

Langkah 3: Hitung matriks adjoin (transpose dari matriks kofaktor).
Adjoin(A) = [[-8, 0, 4], [8, 0, -8], [8, -2, -5]]

Langkah 4: Hitung invers matriks A.
A⁻¹ = (1 / det(A)) * Adjoin(A)
A⁻¹ = (1 / -8) * [[-8, 0, 4], [8, 0, -8], [8, -2, -5]]
A⁻¹ = [[1, 0, -1/2], [-1, 0, 1], [-1, 1/4, 5/8]]

b. Menentukan matriks B jika AB = C:

Kita tahu bahwa A⁻¹C = B.
Sekarang kita kalikan A⁻¹ dengan C:

B = [[1, 0, -1/2], [-1, 0, 1], [-1, 1/4, 5/8]] * [[8, 7, 45], [23, -30, 9], [10, 20, 78]]

Elemen B₁₁ = (1*8) + (0*23) + (-1/2*10) = 8 + 0 - 5 = 3
Elemen B₁₂ = (1*7) + (0*-30) + (-1/2*20) = 7 + 0 - 10 = -3
Elemen B₁₃ = (1*45) + (0*9) + (-1/2*78) = 45 + 0 - 39 = 6

Elemen B₂₁ = (-1*8) + (0*23) + (1*10) = -8 + 0 + 10 = 2
Elemen B₂₂ = (-1*7) + (0*-30) + (1*20) = -7 + 0 + 20 = 13
Elemen B₂₃ = (-1*45) + (0*9) + (1*78) = -45 + 0 + 78 = 33

Elemen B₃₁ = (-1*8) + (1/4*23) + (5/8*10) = -8 + 23/4 + 50/8 = -8 + 23/4 + 25/4 = -8 + 48/4 = -8 + 12 = 4
Elemen B₃₂ = (-1*7) + (1/4*-30) + (5/8*20) = -7 - 30/4 + 100/8 = -7 - 15/2 + 25/2 = -7 + 10/2 = -7 + 5 = -2
Elemen B₃₃ = (-1*45) + (1/4*9) + (5/8*78) = -45 + 9/4 + 390/8 = -45 + 9/4 + 195/4 = -45 + 204/4 = -45 + 51 = 6

Jadi, matriks B adalah:
B = [[3, -3, 6], [2, 13, 33], [4, -2, 6]]