Diketahui matriks P=(5 0 0 6 14 0 6 3 11) dan Q=(-1 0 0 -2

(PQ)^-1 = (QP)^-1 karena PQ = QP.

Pembahasan

Untuk menunjukkan bahwa (PQ)^-1 = (QP)^-1, kita perlu menghitung (PQ)^-1 dan (QP)^-1 secara terpisah dan membandingkannya.

Matriks P:
P = [[5, 0, 0], [6, 14, 0], [6, 3, 11]]

Matriks Q:
Q = [[-1, 0, 0], [-2, -4, 0], [-2, -1, -3]]

Langkah 1: Hitung PQ
PQ = [[5, 0, 0], [6, 14, 0], [6, 3, 11]] * [[-1, 0, 0], [-2, -4, 0], [-2, -1, -3]]

Elemen baris 1:
(5)(-1) + (0)(-2) + (0)(-2) = -5
(5)(0) + (0)(-4) + (0)(-1) = 0
(5)(0) + (0)(0) + (0)(-3) = 0

Elemen baris 2:
(6)(-1) + (14)(-2) + (0)(-2) = -6 - 28 = -34
(6)(0) + (14)(-4) + (0)(-1) = -56
(6)(0) + (14)(0) + (0)(-3) = 0

Elemen baris 3:
(6)(-1) + (3)(-2) + (11)(-2) = -6 - 6 - 22 = -34
(6)(0) + (3)(-4) + (11)(-1) = -12 - 11 = -23
(6)(0) + (3)(0) + (11)(-3) = -33

Jadi, PQ = [[-5, 0, 0], [-34, -56, 0], [-34, -23, -33]]

Langkah 2: Hitung invers dari PQ, yaitu (PQ)^-1
Karena PQ adalah matriks segitiga bawah, inversnya juga akan menjadi matriks segitiga bawah. Kita bisa mencari inversnya menggunakan metode adjoin atau dengan mencari solusi dari PQ * X = I.

Untuk matriks segitiga, determinannya adalah hasil kali elemen diagonalnya.
det(PQ) = (-5)(-56)(-33) = -9240

Menghitung invers secara manual untuk matriks 3x3 cukup rumit. Mari kita gunakan sifat bahwa jika P dan Q adalah matriks invertible, maka (PQ)^-1 = Q^-1 P^-1.

Langkah 3: Hitung QP
QP = [[-1, 0, 0], [-2, -4, 0], [-2, -1, -3]] * [[5, 0, 0], [6, 14, 0], [6, 3, 11]]

Elemen baris 1:
(-1)(5) + (0)(6) + (0)(6) = -5
(-1)(0) + (0)(14) + (0)(3) = 0
(-1)(0) + (0)(0) + (0)(11) = 0

Elemen baris 2:
(-2)(5) + (-4)(6) + (0)(6) = -10 - 24 = -34
(-2)(0) + (-4)(14) + (0)(3) = -56
(-2)(0) + (-4)(0) + (0)(11) = 0

Elemen baris 3:
(-2)(5) + (-1)(6) + (-3)(6) = -10 - 6 - 18 = -34
(-2)(0) + (-1)(14) + (-3)(3) = -14 - 9 = -23
(-2)(0) + (-1)(0) + (-3)(11) = -33

Jadi, QP = [[-5, 0, 0], [-34, -56, 0], [-34, -23, -33]]

Langkah 4: Hitung invers dari QP, yaitu (QP)^-1
Perhatikan bahwa PQ = QP.

Karena PQ = QP, maka inversnya juga akan sama: (PQ)^-1 = (QP)^-1.

Untuk membuktikannya secara formal, kita perlu menghitung invers dari matriks:
M = [[-5, 0, 0], [-34, -56, 0], [-34, -23, -33]]

Determinan M = (-5) * (-56) * (-33) = -9240.

Minor:
M11 = (-56)(-33) - (0)(-23) = 1848
M12 = (-34)(-33) - (0)(-34) = 1122
M13 = (-34)(-23) - (-56)(-34) = 782 - 1904 = -1122

M21 = (0)(-33) - (0)(-23) = 0
M22 = (-5)(-33) - (0)(-34) = 165
M23 = (-5)(-23) - (0)(-34) = 115

M31 = (0)(0) - (0)(-56) = 0
M32 = (-5)(0) - (0)(-34) = 0
M33 = (-5)(-56) - (0)(-34) = 280

Kofaktor:
C11 = 1848
C12 = -1122
C13 = -1122
C21 = 0
C22 = 165
C23 = -115
C31 = 0
C32 = 0
C33 = 280

Adjoin (transpose kofaktor):
adj(M) = [[1848, 0, 0], [-1122, 165, 0], [-1122, -115, 280]]

Invers M = (1/det(M)) * adj(M)
(PQ)^-1 = (QP)^-1 = (1/-9240) * [[1848, 0, 0], [-1122, 165, 0], [-1122, -115, 280]]

(PQ)^-1 = [[1848/-9240, 0, 0], [-1122/-9240, 165/-9240, 0], [-1122/-9240, -115/-9240, 280/-9240]]

(PQ)^-1 = [[-1/5, 0, 0], [11/90, -11/616, 0], [11/90, 23/1848, -1/33]]

Karena PQ = QP, maka inversnya juga sama, yaitu (PQ)^-1 = (QP)^-1.