Misal diberikan matriks A = ( 1 -1 0 -1 0 1 1 1 1) dan B =

(BA)^-1 = [[-1, -2/3, -2/3], [0, -2/3, 1/3], [0, 1/3, 1/3]].

Pembahasan

Diberikan matriks A = [[1, -1, 0], [-1, 0, 1], [1, 1, 1]] dan B = [[-1, -1, -1], [1, 1, 0], [0, 1, 1]]. Kita perlu menentukan (BA)^-1.

Langkah 1: Hitung hasil perkalian matriks BA.
BA = [[-1, -1, -1], [1, 1, 0], [0, 1, 1]] * [[1, -1, 0], [-1, 0, 1], [1, 1, 1]]
BA = [[(-1*1)+(-1*-1)+(-1*1), (-1*-1)+(-1*0)+(-1*1), (-1*0)+(-1*1)+(-1*1)],
      [(1*1)+(1*-1)+(0*1), (1*-1)+(1*0)+(0*1), (1*0)+(1*1)+(0*1)],
      [(0*1)+(1*-1)+(1*1), (0*-1)+(1*0)+(1*1), (0*0)+(1*1)+(1*1)]]
BA = [[-1+1-1, 1+0-1, 0-1-1], [1-1+0, -1+0+0, 0+1+0], [0-1+1, 0+0+1, 0+1+1]]
BA = [[-1, 0, -2], [0, -1, 1], [0, 1, 2]]

Langkah 2: Hitung determinan dari matriks BA.
det(BA) = -1 * ((-1*2) - (1*1)) - 0 * (...) + (-2) * ((0*1) - (-1*0))
det(BA) = -1 * (-2 - 1) - 0 + (-2) * (0)
det(BA) = -1 * (-3) = 3

Langkah 3: Hitung invers dari matriks BA.
(BA)^-1 = (1/det(BA)) * adj(BA)

Untuk mencari adjoint (adj(BA)), kita perlu mencari matriks kofaktor terlebih dahulu.
Kofaktor C11 = (-1)^(1+1) * det([[-1, 1], [1, 2]]) = 1 * (-2 - 1) = -3
Kofaktor C12 = (-1)^(1+2) * det([[0, 1], [0, 2]]) = -1 * (0 - 0) = 0
Kofaktor C13 = (-1)^(1+3) * det([[0, -1], [0, 1]]) = 1 * (0 - 0) = 0

Kofaktor C21 = (-1)^(2+1) * det([[0, -2], [1, 2]]) = -1 * (0 - (-2)) = -2
Kofaktor C22 = (-1)^(2+2) * det([[-1, -2], [0, 2]]) = 1 * (-2 - 0) = -2
Kofaktor C23 = (-1)^(2+3) * det([[-1, 0], [0, 1]]) = -1 * (-1 - 0) = 1

Kofaktor C31 = (-1)^(3+1) * det([[0, -2], [-1, 1]]) = 1 * (0 - 2) = -2
Kofaktor C32 = (-1)^(3+2) * det([[-1, -2], [0, 1]]) = -1 * (-1 - 0) = 1
Kofaktor C33 = (-1)^(3+3) * det([[-1, 0], [0, -1]]) = 1 * (1 - 0) = 1

Matriks Kofaktor = [[-3, 0, 0], [-2, -2, 1], [-2, 1, 1]]

Adjoint (BA) = transpose dari Matriks Kofaktor = [[-3, -2, -2], [0, -2, 1], [0, 1, 1]]

(BA)^-1 = (1/3) * [[-3, -2, -2], [0, -2, 1], [0, 1, 1]]
(BA)^-1 = [[-1, -2/3, -2/3], [0, -2/3, 1/3], [0, 1/3, 1/3]]

Jadi, (BA)^-1 adalah [[-1, -2/3, -2/3], [0, -2/3, 1/3], [0, 1/3, 1/3]].