Tentukan invers matriks berikut. B = (2 1 1 -2 3 0 3 5 2)

Invers matriks B adalah [[-2, -1, 1], [-4/3, -1/3, 2/3], [19/3, 7/3, -8/3]].

Pembahasan

Untuk menentukan invers matriks B = [[2, 1, 1], [-2, 3, 0], [3, 5, 2]], kita perlu mengikuti langkah-langkah berikut:

1.  **Hitung Determinan (det(B))**:
    det(B) = 2 * ((3*2) - (0*5)) - 1 * ((-2*2) - (0*3)) + 1 * ((-2*5) - (3*3))
    det(B) = 2 * (6 - 0) - 1 * (-4 - 0) + 1 * (-10 - 9)
    det(B) = 2 * 6 - 1 * (-4) + 1 * (-19)
    det(B) = 12 + 4 - 19
    det(B) = -3
    Karena determinan tidak sama dengan nol, matriks B memiliki invers.

2.  **Hitung Matriks Adjoin (adj(B))**:
    Matriks adjoin diperoleh dari transpose matriks kofaktor. Pertama, kita hitung matriks kofaktornya.
    Kofaktor C11 = (-1)^(1+1) * det([[3, 0], [5, 2]]) = 1 * (6 - 0) = 6
    Kofaktor C12 = (-1)^(1+2) * det([[-2, 0], [3, 2]]) = -1 * (-4 - 0) = 4
    Kofaktor C13 = (-1)^(1+3) * det([[-2, 3], [3, 5]]) = 1 * (-10 - 9) = -19
    Kofaktor C21 = (-1)^(2+1) * det([[1, 1], [5, 2]]) = -1 * (2 - 5) = 3
    Kofaktor C22 = (-1)^(2+2) * det([[2, 1], [3, 2]]) = 1 * (4 - 3) = 1
    Kofaktor C23 = (-1)^(2+3) * det([[2, 1], [3, 5]]) = -1 * (10 - 3) = -7
    Kofaktor C31 = (-1)^(3+1) * det([[1, 1], [3, 0]]) = 1 * (0 - 3) = -3
    Kofaktor C32 = (-1)^(3+2) * det([[2, 1], [-2, 0]]) = -1 * (0 - (-2)) = -2
    Kofaktor C33 = (-1)^(3+3) * det([[2, 1], [-2, 3]]) = 1 * (6 - (-2)) = 8

    Matriks Kofaktor = [[6, 4, -19], [3, 1, -7], [-3, -2, 8]]

    Matriks Adjoin (adj(B)) = Transpose dari Matriks Kofaktor
    adj(B) = [[6, 3, -3], [4, 1, -2], [-19, -7, 8]]

3.  **Hitung Invers (B^-1)**:
    B^-1 = (1 / det(B)) * adj(B)
    B^-1 = (1 / -3) * [[6, 3, -3], [4, 1, -2], [-19, -7, 8]]
    B^-1 = [[-2, -1, 1], [-4/3, -1/3, 2/3], [19/3, 7/3, -8/3]]