Diketahui A = (10 -8 8 -6) dan B = (4 6 2 4). Tentukan a.(A

Hasil perhitungan invers matriks A.B dan perkalian invers matriks B dengan A adalah sama, yaitu [[3/2, -7/4], [-5/4, 3/2]].

Pembahasan

Untuk menentukan a.(A . B)^-1 dan b. B^-1 A^-1, kita perlu melakukan langkah-langkah berikut:

1.  **Hitung A . B (Perkalian Matriks A dengan B):**
    A = [[10, -8], [8, -6]]
    B = [[4, 6], [2, 4]]
    A . B = [[(10*4 + -8*2), (10*6 + -8*4)], [(8*4 + -6*2), (8*6 + -6*4)]]
    A . B = [[(40 - 16), (60 - 32)], [(32 - 12), (48 - 24)]]
    A . B = [[24, 28], [20, 24]]

2.  **Hitung Determinan (A . B):**
    det(A . B) = (24 * 24) - (28 * 20)
    det(A . B) = 576 - 560
    det(A . B) = 16

3.  **Hitung Invers (A . B)^-1:**
    (A . B)^-1 = 1/det(A . B) * [[adjugate dari (A . B)]]
    (A . B)^-1 = 1/16 * [[24, -28], [-20, 24]]
    (A . B)^-1 = [[24/16, -28/16], [-20/16, 24/16]]
    (A . B)^-1 = [[3/2, -7/4], [-5/4, 3/2]]

4.  **Hitung Invers B (B^-1):**
    det(B) = (4 * 4) - (6 * 2)
    det(B) = 16 - 12
    det(B) = 4
    B^-1 = 1/det(B) * [[adjugate dari B]]
    B^-1 = 1/4 * [[4, -6], [-2, 4]]
    B^-1 = [[4/4, -6/4], [-2/4, 4/4]]
    B^-1 = [[1, -3/2], [-1/2, 1]]

5.  **Hitung Invers A (A^-1):**
    det(A) = (10 * -6) - (-8 * 8)
    det(A) = -60 - (-64)
    det(A) = -60 + 64
    det(A) = 4
    A^-1 = 1/det(A) * [[adjugate dari A]]
    A^-1 = 1/4 * [[-6, 8], [-8, 10]]
    A^-1 = [[-6/4, 8/4], [-8/4, 10/4]]
    A^-1 = [[-3/2, 2], [-2, 5/2]]

6.  **Hitung B^-1 A^-1:**
    B^-1 A^-1 = [[1, -3/2], [-1/2, 1]] * [[-3/2, 2], [-2, 5/2]]
    B^-1 A^-1 = [[(1*-3/2 + -3/2*-2), (1*2 + -3/2*5/2)], [(-1/2*-3/2 + 1*-2), (-1/2*2 + 1*5/2)]]
    B^-1 A^-1 = [[(-3/2 + 3), (2 - 15/4)], [(3/4 - 2), (-1 + 5/2)]]
    B^-1 A^-1 = [[3/2, -7/4], [-5/4, 3/2]]

**Jawaban:**
a. (A . B)^-1 = [[3/2, -7/4], [-5/4, 3/2]]
b. B^-1 A^-1 = [[3/2, -7/4], [-5/4, 3/2]]