Diketahui matriks A=(-1 0 3 0 2 1 -3 1 0) B=(-5 -1 3 -3 0

Invers matriks A adalah 1/19 * [[-1, 3, -4], [-3, 9, 7], [6, 1, -2]] dan matriks C adalah [[24/19, 13/19, -4/19], [-61/19, -18/19, -50/19], [-1, 0, 1]].

Pembahasan

Untuk menentukan invers matriks A, kita perlu menggunakan rumus:
A<sup>-1</sup> = 1/det(A) * adj(A)

Pertama, hitung determinan matriks A:
det(A) = -1 * (2*0 - 1*1) - 0 * (...) + 3 * (0*1 - 2*(-3))
det(A) = -1 * (-1) - 0 + 3 * (6)
det(A) = 1 + 18
det(A) = 19

Selanjutnya, hitung matriks adjoin dari A (dengan menukar elemen diagonal utama, mengubah tanda elemen diagonal sekunder, dan melakukan transpos):

Matriks Kofaktor:
C<sub>11</sub> = (2*0 - 1*1) = -1
C<sub>12</sub> = -(0*0 - 1*(-3)) = -3
C<sub>13</sub> = (0*1 - 2*(-3)) = 6
C<sub>21</sub> = -(0*0 - 3*1) = 3
C<sub>22</sub> = (-1*0 - 3*(-3)) = 9
C<sub>23</sub> = -(-1*1 - 0*(-3)) = 1
C<sub>31</sub> = (0*1 - 2*2) = -4
C<sub>32</sub> = -(-1*1 - 3*2) = -(-1 - 6) = 7
C<sub>33</sub> = (-1*2 - 0*0) = -2

Matriks Kofaktor = [[-1, -3, 6], [3, 9, 1], [-4, 7, -2]]

Adjoin(A) = Transpose(Matriks Kofaktor) = [[-1, 3, -4], [-3, 9, 7], [6, 1, -2]]

Invers Matriks A (A<sup>-1</sup>) = 1/19 * [[-1, 3, -4], [-3, 9, 7], [6, 1, -2]]

Untuk menentukan matriks C, kita gunakan rumus AC = B, maka C = A<sup>-1</sup>B:
C = 1/19 * [[-1, 3, -4], [-3, 9, 7], [6, 1, -2]] * [[-5, -1, 3], [-3, 0, -3], [-7, -3, -2]]

Hitung perkalian matriks:

Baris 1 C:
C<sub>11</sub> = 1/19 * (-1*(-5) + 3*(-3) + (-4)*(-7)) = 1/19 * (5 - 9 + 28) = 24/19
C<sub>12</sub> = 1/19 * (-1*(-1) + 3*0 + (-4)*(-3)) = 1/19 * (1 + 0 + 12) = 13/19
C<sub>13</sub> = 1/19 * (-1*3 + 3*(-3) + (-4)*(-2)) = 1/19 * (-3 - 9 + 8) = -4/19

Baris 2 C:
C<sub>21</sub> = 1/19 * (-3*(-5) + 9*(-3) + 7*(-7)) = 1/19 * (15 - 27 - 49) = -61/19
C<sub>22</sub> = 1/19 * (-3*(-1) + 9*0 + 7*(-3)) = 1/19 * (3 + 0 - 21) = -18/19
C<sub>23</sub> = 1/19 * (-3*3 + 9*(-3) + 7*(-2)) = 1/19 * (-9 - 27 - 14) = -50/19

Baris 3 C:
C<sub>31</sub> = 1/19 * (6*(-5) + 1*(-3) + (-2)*(-7)) = 1/19 * (-30 - 3 + 14) = -19/19 = -1
C<sub>32</sub> = 1/19 * (6*(-1) + 1*0 + (-2)*(-3)) = 1/19 * (-6 + 0 + 6) = 0/19 = 0
C<sub>33</sub> = 1/19 * (6*3 + 1*(-3) + (-2)*(-2)) = 1/19 * (18 - 3 + 4) = 19/19 = 1

Jadi, matriks C adalah:
C = [[24/19, 13/19, -4/19], [-61/19, -18/19, -50/19], [-1, 0, 1]]