Diketahui matriks A=(-1 2 3 -1 -1 3 -2 5 1). Jika matriks

Determinan matriks B adalah 225.

Pembahasan

Diberikan matriks A = [[-1, 2, 3], [-1, -1, 3], [-2, 5, 1]].
Kita perlu mencari determinan dari matriks B = A^2.

Langkah 1: Hitung matriks A^2.
A^2 = A * A
    = [[-1, 2, 3], [-1, -1, 3], [-2, 5, 1]] * [[-1, 2, 3], [-1, -1, 3], [-2, 5, 1]]

Elemen B[1,1] = (-1)(-1) + (2)(-1) + (3)(-2) = 1 - 2 - 6 = -7
Elemen B[1,2] = (-1)(2) + (2)(-1) + (3)(5) = -2 - 2 + 15 = 11
Elemen B[1,3] = (-1)(3) + (2)(3) + (3)(1) = -3 + 6 + 3 = 6

Elemen B[2,1] = (-1)(-1) + (-1)(-1) + (3)(-2) = 1 + 1 - 6 = -4
Elemen B[2,2] = (-1)(2) + (-1)(-1) + (3)(5) = -2 + 1 + 15 = 14
Elemen B[2,3] = (-1)(3) + (-1)(3) + (3)(1) = -3 - 3 + 3 = -3

Elemen B[3,1] = (-2)(-1) + (5)(-1) + (1)(-2) = 2 - 5 - 2 = -5
Elemen B[3,2] = (-2)(2) + (5)(-1) + (1)(5) = -4 - 5 + 5 = -4
Elemen B[3,3] = (-2)(3) + (5)(3) + (1)(1) = -6 + 15 + 1 = 10

Jadi, matriks B = A^2 = [[-7, 11, 6], [-4, 14, -3], [-5, -4, 10]].

Langkah 2: Hitung determinan matriks B.
det(B) = -7 * det([[14, -3], [-4, 10]]) - 11 * det([[-4, -3], [-5, 10]]) + 6 * det([[-4, 14], [-5, -4]])

det(B) = -7 * ((14)(10) - (-3)(-4)) - 11 * ((-4)(10) - (-3)(-5)) + 6 * ((-4)(-4) - (14)(-5))
det(B) = -7 * (140 - 12) - 11 * (-40 - 15) + 6 * (16 - (-70))
det(B) = -7 * (128) - 11 * (-55) + 6 * (16 + 70)
det(B) = -896 + 605 + 6 * 86
det(B) = -896 + 605 + 516
det(B) = -896 + 1121
det(B) = 225

Alternatif menggunakan sifat determinan: det(A^2) = (det(A))^2.
Hitung determinan matriks A:
det(A) = -1 * det([[-1, 3], [5, 1]]) - 2 * det([[-1, 3], [-2, 1]]) + 3 * det([[-1, -1], [-2, 5]])
    = -1 * ((-1)(1) - (3)(5)) - 2 * ((-1)(1) - (3)(-2)) + 3 * ((-1)(5) - (-1)(-2))
    = -1 * (-1 - 15) - 2 * (-1 - (-6)) + 3 * (-5 - 2)
    = -1 * (-16) - 2 * (-1 + 6) + 3 * (-7)
    = 16 - 2 * (5) - 21
    = 16 - 10 - 21
    = 6 - 21
    = -15

Jadi, det(B) = det(A^2) = (det(A))^2 = (-15)^2 = 225.