Diketahui matriks A=[1 1 2 0 2 1 1 0 2] dan A^3=[10 9 23 5

A^2=[[3, 3, 7], [1, 4, 4], [3, 1, 6]]. A^3 - 5A^2 + 6A - I = O terbukti. A(A-2I)(A-3I) = I terbukti.

Pembahasan

Diketahui matriks A=[[1, 1, 2], [0, 2, 1], [1, 0, 2]] dan A^3=[[10, 9, 23], [5, 9, 14], [9, 5, 19]].

a. Menentukan matriks A^2:
A^2 = A * A = [[1, 1, 2], [0, 2, 1], [1, 0, 2]] * [[1, 1, 2], [0, 2, 1], [1, 0, 2]]
   = [[(1*1)+(1*0)+(2*1), (1*1)+(1*2)+(2*0), (1*2)+(1*1)+(2*2)],
      [(0*1)+(2*0)+(1*1), (0*1)+(2*2)+(1*0), (0*2)+(2*1)+(1*2)],
      [(1*1)+(0*0)+(2*1), (1*1)+(0*2)+(2*0), (1*2)+(0*1)+(2*2)]]
   = [[1+0+2, 1+2+0, 2+1+4], [0+0+1, 0+4+0, 0+2+2], [1+0+2, 1+0+0, 2+0+4]]
   = [[3, 3, 7], [1, 4, 4], [3, 1, 6]]

b. Menunjukkan bahwa A^3 - 5A^2 + 6A - I = O:
Kita perlu menghitung 5A^2, 6A, dan I (matriks identitas).
5A^2 = 5 * [[3, 3, 7], [1, 4, 4], [3, 1, 6]] = [[15, 15, 35], [5, 20, 20], [15, 5, 30]]
6A = 6 * [[1, 1, 2], [0, 2, 1], [1, 0, 2]] = [[6, 6, 12], [0, 12, 6], [6, 0, 12]]
I = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

A^3 - 5A^2 + 6A - I =
[[10, 9, 23], [5, 9, 14], [9, 5, 19]] - [[15, 15, 35], [5, 20, 20], [15, 5, 30]] + [[6, 6, 12], [0, 12, 6], [6, 0, 12]] - [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
=
[[(10-15+6-1), (9-15+6-0), (23-35+12-0)],
 [(5-5+0-0), (9-20+12-1), (14-20+6-0)],
 [(9-15+6-0), (5-5+0-0), (19-30+12-1)]]
=
[[0, 0, 0], [0, 0, 0], [0, 0, 0]]
Ini menunjukkan bahwa A^3 - 5A^2 + 6A - I = O (matriks nol).

c. Menunjukkan bahwa A(A-2I)(A-3I) = I:
Pertama, hitung (A-2I) dan (A-3I):
A-2I = [[1, 1, 2], [0, 2, 1], [1, 0, 2]] - [[2, 0, 0], [0, 2, 0], [0, 0, 2]] = [[-1, 1, 2], [0, 0, 1], [1, 0, 0]]
A-3I = [[1, 1, 2], [0, 2, 1], [1, 0, 2]] - [[3, 0, 0], [0, 3, 0], [0, 0, 3]] = [[-2, 1, 2], [0, -1, 1], [1, 0, -1]]

Selanjutnya, hitung (A-2I)(A-3I):
(A-2I)(A-3I) = [[-1, 1, 2], [0, 0, 1], [1, 0, 0]] * [[-2, 1, 2], [0, -1, 1], [1, 0, -1]]
             = [[((-1)*(-2))+(1*0)+(2*1), ((-1)*1)+(1*(-1))+(2*0), ((-1)*2)+(1*1)+(2*(-1)))],
               [((0)*(-2))+(0*0)+(1*1), (0*1)+(0*(-1))+(1*0), (0*2)+(0*1)+(1*(-1)))],
               [((1)*(-2))+(0*0)+(0*1), (1*1)+(0*(-1))+(0*0), (1*2)+(0*1)+(0*(-1)))]]
             = [[(2+0+2), (-1-1+0), (-2+1-2)], [0+0+1, 0+0+0, 0+0-1], [-2+0+0, 1+0+0, 2+0+0]]
             = [[4, -2, -3], [1, 0, -1], [-2, 1, 2]]

Terakhir, hitung A * [(A-2I)(A-3I)]:
A * [[4, -2, -3], [1, 0, -1], [-2, 1, 2]]
= [[1, 1, 2], [0, 2, 1], [1, 0, 2]] * [[4, -2, -3], [1, 0, -1], [-2, 1, 2]]
= [[(1*4)+(1*1)+(2*(-2)), (1*(-2))+(1*0)+(2*1), (1*(-3))+(1*(-1))+(2*2)],
   [(0*4)+(2*1)+(1*(-2)), (0*(-2))+(2*0)+(1*1), (0*(-3))+(2*(-1))+(1*2)],
   [(1*4)+(0*1)+(2*(-2)), (1*(-2))+(0*0)+(2*1), (1*(-3))+(0*(-1))+(2*2)]]
= [[(4+1-4), (-2+0+2), (-3-1+4)], [0+2-2, 0+0+1, 0-2+2], [4+0-4, -2+0+2, -3+0+4]]
= [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
Ini menunjukkan bahwa A(A-2I)(A-3I) = I (matriks identitas).