Diketahui matriks A = [1 -1 3 1 2 -1 3 1 -2] Tentukan: a.

Minor matriks A adalah M11=-3, M12=1, M13=-5, M21=-1, M22=-11, M23=4, M31=-5, M32=-4, M33=3. Kofaktor matriks A adalah C11=-3, C12=-1, C13=-5, C21=1, C22=-11, C23=-4, C31=-5, C32=4, C33=3. Adjoin A = [[-3, 1, -5], [-1, -11, 4], [-5, -4, 3]].

Pembahasan

Matriks A adalah matriks 3x3 yang elemennya adalah A = [[1, -1, 3], [1, 2, -1], [3, 1, -2]].

a. Minor matriks A:
Minor M_ij adalah determinan dari submatriks yang diperoleh dengan menghapus baris ke-i dan kolom ke-j.

M_11 = det([[2, -1], [1, -2]]) = (2)(-2) - (-1)(1) = -4 + 1 = -3
M_12 = det([[1, -1], [3, -2]]) = (1)(-2) - (-1)(3) = -2 + 3 = 1
M_13 = det([[1, 2], [3, 1]]) = (1)(1) - (2)(3) = 1 - 6 = -5
M_21 = det([[-1, 3], [1, -2]]) = (-1)(-2) - (3)(1) = 2 - 3 = -1
M_22 = det([[1, 3], [3, -2]]) = (1)(-2) - (3)(3) = -2 - 9 = -11
M_23 = det([[1, -1], [3, 1]]) = (1)(1) - (-1)(3) = 1 + 3 = 4
M_31 = det([[-1, 3], [2, -1]]) = (-1)(-1) - (3)(2) = 1 - 6 = -5
M_32 = det([[1, 3], [1, -1]]) = (1)(-1) - (3)(1) = -1 - 3 = -4
M_33 = det([[1, -1], [1, 2]]) = (1)(2) - (-1)(1) = 2 + 1 = 3

b. Kofaktor matriks A:
Kofaktor C_ij = (-1)^(i+j) * M_ij.

C_11 = (-1)^(1+1) * M_11 = 1 * (-3) = -3
C_12 = (-1)^(1+2) * M_12 = -1 * (1) = -1
C_13 = (-1)^(1+3) * M_13 = 1 * (-5) = -5
C_21 = (-1)^(2+1) * M_21 = -1 * (-1) = 1
C_22 = (-1)^(2+2) * M_22 = 1 * (-11) = -11
C_23 = (-1)^(2+3) * M_23 = -1 * (4) = -4
C_31 = (-1)^(3+1) * M_31 = 1 * (-5) = -5
C_32 = (-1)^(3+2) * M_32 = -1 * (-4) = 4
C_33 = (-1)^(3+3) * M_33 = 1 * (3) = 3

c. Adjoin A:
Adjoin A adalah transpose dari matriks kofaktor.
Adjoin A = C^T = [[-3, 1, -5], [-1, -11, 4], [-5, -4, 3]]