Tunjukkan bahwa: (-1 -1 -1 0 1 0 0 0 1)^2=(0 1 0 -1 -1 -1 0

Terbukti bahwa hasil pemangkatan matriks-matriks tersebut menghasilkan matriks identitas (I) setelah dilakukan perhitungan perkalian matriks.

Pembahasan

Soal ini meminta kita untuk menunjukkan bahwa hasil pemangkatan matriks tertentu menghasilkan matriks identitas (I). Diberikan tiga ekspresi matriks:

M = (-1 -1 -1 0 1 0 0 0 1)
N = (0 1 0 -1 -1 -1 0 0 1)
O = (0 1 0 0 0 1 -1 -1 -1)

Kita perlu menunjukkan bahwa M^2 = I, N^3 = I, dan O^4 = I.

Mari kita hitung M^2:
M^2 = M * M

Matriks M dapat ditulis dalam bentuk:
M = [[-1, -1, -1], [0, 1, 0], [0, 0, 1]]

M^2 = [[-1, -1, -1], [0, 1, 0], [0, 0, 1]] * [[-1, -1, -1], [0, 1, 0], [0, 0, 1]]

Untuk menghitung M^2, kita lakukan perkalian matriks:
Elemen (1,1) = (-1*-1) + (-1*0) + (-1*0) = 1 + 0 + 0 = 1
Elemen (1,2) = (-1*-1) + (-1*1) + (-1*0) = 1 - 1 + 0 = 0
Elemen (1,3) = (-1*-1) + (-1*0) + (-1*1) = 1 + 0 - 1 = 0

Elemen (2,1) = (0*-1) + (1*0) + (0*0) = 0 + 0 + 0 = 0
Elemen (2,2) = (0*-1) + (1*1) + (0*0) = 0 + 1 + 0 = 1
Elemen (2,3) = (0*-1) + (1*0) + (0*1) = 0 + 0 + 0 = 0

Elemen (3,1) = (0*-1) + (0*0) + (1*0) = 0 + 0 + 0 = 0
Elemen (3,2) = (0*-1) + (0*1) + (1*0) = 0 + 0 + 0 = 0
Elemen (3,3) = (0*-1) + (0*0) + (1*1) = 0 + 0 + 1 = 1

Jadi, M^2 = [[1, 0, 0], [0, 1, 0], [0, 0, 1]] = I. Pernyataan pertama terbukti.

Selanjutnya, mari kita hitung N^3:
N = [[0, 1, 0], [-1, -1, -1], [0, 0, 1]]

N^2 = N * N
N^2 = [[0, 1, 0], [-1, -1, -1], [0, 0, 1]] * [[0, 1, 0], [-1, -1, -1], [0, 0, 1]]

Elemen (1,1) = (0*0) + (1*-1) + (0*0) = 0 - 1 + 0 = -1
Elemen (1,2) = (0*1) + (1*-1) + (0*0) = 0 - 1 + 0 = -1
Elemen (1,3) = (0*0) + (1*-1) + (0*1) = 0 - 1 + 0 = -1

Elemen (2,1) = (-1*0) + (-1*-1) + (-1*0) = 0 + 1 + 0 = 1
Elemen (2,2) = (-1*1) + (-1*-1) + (-1*0) = -1 + 1 + 0 = 0
Elemen (2,3) = (-1*0) + (-1*-1) + (-1*1) = 0 + 1 - 1 = 0

Elemen (3,1) = (0*0) + (0*-1) + (1*0) = 0 + 0 + 0 = 0
Elemen (3,2) = (0*1) + (0*-1) + (1*0) = 0 + 0 + 0 = 0
Elemen (3,3) = (0*0) + (0*-1) + (1*1) = 0 + 0 + 1 = 1

Jadi, N^2 = [[-1, -1, -1], [1, 0, 0], [0, 0, 1]].

Sekarang hitung N^3 = N^2 * N:
N^3 = [[-1, -1, -1], [1, 0, 0], [0, 0, 1]] * [[0, 1, 0], [-1, -1, -1], [0, 0, 1]]

Elemen (1,1) = (-1*0) + (-1*-1) + (-1*0) = 0 + 1 + 0 = 1
Elemen (1,2) = (-1*1) + (-1*-1) + (-1*0) = -1 + 1 + 0 = 0
Elemen (1,3) = (-1*0) + (-1*-1) + (-1*1) = 0 + 1 - 1 = 0

Elemen (2,1) = (1*0) + (0*-1) + (0*0) = 0 + 0 + 0 = 0
Elemen (2,2) = (1*1) + (0*-1) + (0*0) = 1 + 0 + 0 = 1
Elemen (2,3) = (1*0) + (0*-1) + (0*1) = 0 + 0 + 0 = 0

Elemen (3,1) = (0*0) + (0*-1) + (1*0) = 0 + 0 + 0 = 0
Elemen (3,2) = (0*1) + (0*-1) + (1*0) = 0 + 0 + 0 = 0
Elemen (3,3) = (0*0) + (0*-1) + (1*1) = 0 + 0 + 1 = 1

Jadi, N^3 = [[1, 0, 0], [0, 1, 0], [0, 0, 1]] = I. Pernyataan kedua terbukti.

Terakhir, mari kita hitung O^4:
O = [[0, 1, 0], [0, 0, 1], [-1, -1, -1]]

O^2 = O * O
O^2 = [[0, 1, 0], [0, 0, 1], [-1, -1, -1]] * [[0, 1, 0], [0, 0, 1], [-1, -1, -1]]

Elemen (1,1) = (0*0) + (1*0) + (0*-1) = 0 + 0 + 0 = 0
Elemen (1,2) = (0*1) + (1*0) + (0*-1) = 0 + 0 + 0 = 0
Elemen (1,3) = (0*0) + (1*1) + (0*-1) = 0 + 1 + 0 = 1

Elemen (2,1) = (0*0) + (0*0) + (1*-1) = 0 + 0 - 1 = -1
Elemen (2,2) = (0*1) + (0*0) + (1*-1) = 0 + 0 - 1 = -1
Elemen (2,3) = (0*0) + (0*1) + (1*-1) = 0 + 0 - 1 = -1

Elemen (3,1) = (-1*0) + (-1*0) + (-1*-1) = 0 + 0 + 1 = 1
Elemen (3,2) = (-1*1) + (-1*0) + (-1*-1) = -1 + 0 + 1 = 0
Elemen (3,3) = (-1*0) + (-1*1) + (-1*-1) = 0 - 1 + 1 = 0

Jadi, O^2 = [[0, 0, 1], [-1, -1, -1], [1, 0, 0]].

Sekarang hitung O^3 = O^2 * O:
O^3 = [[0, 0, 1], [-1, -1, -1], [1, 0, 0]] * [[0, 1, 0], [0, 0, 1], [-1, -1, -1]]

Elemen (1,1) = (0*0) + (0*0) + (1*-1) = 0 + 0 - 1 = -1
Elemen (1,2) = (0*1) + (0*0) + (1*-1) = 0 + 0 - 1 = -1
Elemen (1,3) = (0*0) + (0*1) + (1*-1) = 0 + 0 - 1 = -1

Elemen (2,1) = (-1*0) + (-1*0) + (-1*-1) = 0 + 0 + 1 = 1
Elemen (2,2) = (-1*1) + (-1*0) + (-1*-1) = -1 + 0 + 1 = 0
Elemen (2,3) = (-1*0) + (-1*1) + (-1*-1) = 0 - 1 + 1 = 0

Elemen (3,1) = (1*0) + (0*0) + (0*-1) = 0 + 0 + 0 = 0
Elemen (3,2) = (1*1) + (0*0) + (0*-1) = 1 + 0 + 0 = 1
Elemen (3,3) = (1*0) + (0*1) + (0*-1) = 0 + 0 + 0 = 0

Jadi, O^3 = [[-1, -1, -1], [1, 0, 0], [0, 1, 0]].

Terakhir, hitung O^4 = O^3 * O:
O^4 = [[-1, -1, -1], [1, 0, 0], [0, 1, 0]] * [[0, 1, 0], [0, 0, 1], [-1, -1, -1]]

Elemen (1,1) = (-1*0) + (-1*0) + (-1*-1) = 0 + 0 + 1 = 1
Elemen (1,2) = (-1*1) + (-1*0) + (-1*-1) = -1 + 0 + 1 = 0
Elemen (1,3) = (-1*0) + (-1*1) + (-1*-1) = 0 - 1 + 1 = 0

Elemen (2,1) = (1*0) + (0*0) + (0*-1) = 0 + 0 + 0 = 0
Elemen (2,2) = (1*1) + (0*0) + (0*-1) = 1 + 0 + 0 = 1
Elemen (2,3) = (1*0) + (0*1) + (0*-1) = 0 + 0 + 0 = 0

Elemen (3,1) = (0*0) + (1*0) + (0*-1) = 0 + 0 + 0 = 0
Elemen (3,2) = (0*1) + (1*0) + (0*-1) = 0 + 0 + 0 = 0
Elemen (3,3) = (0*0) + (1*1) + (0*-1) = 0 + 1 + 0 = 1

Jadi, O^4 = [[1, 0, 0], [0, 1, 0], [0, 0, 1]] = I. Pernyataan ketiga terbukti.

Dengan demikian, terbukti bahwa M^2 = I, N^3 = I, dan O^4 = I.