Tentukan invers dari matriks P=[-3 2 -2 1 0 1 1 -3 0].

Invers matriks P adalah [[-3, -6, -2], [-1, -2, -1], [3, 7, 2]].

Pembahasan

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

1.  **Hitung Determinan Matriks (det(P))**:
    det(P) = -3 * ( (0*0) - (1*-3) ) - 2 * ( (1*0) - (1*1) ) + (-2) * ( (1*-3) - (0*1) )
    det(P) = -3 * (0 - (-3)) - 2 * (0 - 1) + (-2) * (-3 - 0)
    det(P) = -3 * (3) - 2 * (-1) + (-2) * (-3)
    det(P) = -9 + 2 + 6
    det(P) = -1

2.  **Cari Matriks Adjoin (adj(P))**:
    Matriks Adjoin diperoleh dari transpose matriks kofaktor.
    
    **a. Hitung Matriks Minor**: 
    M11 = | 0  1 | = 0*0 - 1*(-3) = 3
         | -3 0 |
    M12 = | 1  1 | = 1*0 - 1*1 = -1
         | 1  0 |
    M13 = | 1  0 | = 1*(-3) - 0*1 = -3
         | 1 -3 |
    M21 = | 2 -2 | = 2*0 - (-2)*(-3) = -6
         | -3 0 |
    M22 = |-3 -2 | = -3*0 - (-2)*1 = 2
         | 1  0 |
    M23 = |-3  2 | = -3*(-3) - 2*1 = 9 - 2 = 7
         | 1 -3 |
    M31 = | 2 -2 | = 2*1 - (-2)*0 = 2
         | 0  1 |
    M32 = |-3 -2 | = -3*1 - (-2)*1 = -3 + 2 = -1
         | 1  1 |
    M33 = |-3  2 | = -3*0 - 2*1 = -2
         | 1  0 |
    
    **b. Hitung Matriks Kofaktor (C)**:
    C11 = +M11 = 3
    C12 = -M12 = -(-1) = 1
    C13 = +M13 = -3
    C21 = -M21 = -(-6) = 6
    C22 = +M22 = 2
    C23 = -M23 = -7
    C31 = +M31 = 2
    C32 = -M32 = -(-1) = 1
    C33 = +M33 = -2
    
    Matriks Kofaktor C = [[3, 1, -3], [6, 2, -7], [2, 1, -2]]
    
    **c. Transpose Matriks Kofaktor untuk mendapatkan Adjoin (adj(P))**:
    adj(P) = C^T = [[3, 6, 2], [1, 2, 1], [-3, -7, -2]]

3.  **Hitung Invers Matriks (P^-1)**:
    P^-1 = (1 / det(P)) * adj(P)
    P^-1 = (1 / -1) * [[3, 6, 2], [1, 2, 1], [-3, -7, -2]]
    P^-1 = -1 * [[3, 6, 2], [1, 2, 1], [-3, -7, -2]]
    P^-1 = [[-3, -6, -2], [-1, -2, -1], [3, 7, 2]]

Jadi, invers dari matriks P adalah [[-3, -6, -2], [-1, -2, -1], [3, 7, 2]].