Tentukanlah invers matriks A=[3 2 4 2 1 3 0 1 -2]

A^-1 = [[-5, 8, 2], [4, -6, -1], [2, -3, -1]]

Pembahasan

Untuk menentukan invers dari matriks A = [3 2 4; 2 1 3; 0 1 -2], kita bisa menggunakan metode adjoin atau metode OBE (Operasi Baris Elementer).

Metode Adjoin:
1. Hitung determinan matriks A (det(A)).
   det(A) = 3 * (1*(-2) - 3*1) - 2 * (2*(-2) - 3*0) + 4 * (2*1 - 1*0)
   det(A) = 3 * (-2 - 3) - 2 * (-4 - 0) + 4 * (2 - 0)
   det(A) = 3 * (-5) - 2 * (-4) + 4 * (2)
   det(A) = -15 + 8 + 8
   det(A) = 1

2. Cari matriks kofaktor (C).
   C11 = +(1*(-2) - 3*1) = -5
   C12 = -(2*(-2) - 3*0) = -(-4) = 4
   C13 = +(2*1 - 1*0) = 2
   C21 = -(2*(-2) - 4*1) = -(-4 - 4) = 8
   C22 = +(3*(-2) - 4*0) = -6
   C23 = -(3*1 - 2*0) = -3
   C31 = +(2*3 - 4*1) = 6 - 4 = 2
   C32 = -(3*3 - 4*2) = -(9 - 8) = -1
   C33 = +(3*1 - 2*2) = 3 - 4 = -1
   
   Matriks Kofaktor C = [[-5, 4, 2], [8, -6, -3], [2, -1, -1]]

3. Cari matriks adjoin (Adj(A)), yaitu transpose dari matriks kofaktor.
   Adj(A) = C^T = [[-5, 8, 2], [4, -6, -1], [2, -3, -1]]

4. Hitung invers matriks A (A^-1) dengan rumus A^-1 = (1/det(A)) * Adj(A).
   A^-1 = (1/1) * [[-5, 8, 2], [4, -6, -1], [2, -3, -1]]
   A^-1 = [[-5, 8, 2], [4, -6, -1], [2, -3, -1]]

Jadi, invers matriks A adalah [[-5, 8, 2], [4, -6, -1], [2, -3, -1]].