Diketahui matriks P=(1 -1 2 3). a. Tentukan hasil P^2, P^3,

Hasil P²=[[-1, -4], [8, 7]], P³=[[-9, -11], [22, 13]], P⁴=[[-31, -24], [48, 17]]. Hasil P²-P³+P⁴=[[-23, -17], [34, 11]].

Pembahasan

Diberikan matriks P = [[1, -1], [2, 3]].
Untuk a. Menentukan hasil P², P³, dan P⁴:
   P² = P * P = [[1, -1], [2, 3]] * [[1, -1], [2, 3]] = [[(1*1)+(-1*2), (1*-1)+(-1*3)], [(2*1)+(3*2), (2*-1)+(3*3)]] = [[1-2, -1-3], [2+6, -2+9]] = [[-1, -4], [8, 7]].
   P³ = P² * P = [[-1, -4], [8, 7]] * [[1, -1], [2, 3]] = [[(-1*1)+(-4*2), (-1*-1)+(-4*3)], [(8*1)+(7*2), (8*-1)+(7*3)]] = [[-1-8, 1-12], [8+14, -8+21]] = [[-9, -11], [22, 13]].
   P⁴ = P³ * P = [[-9, -11], [22, 13]] * [[1, -1], [2, 3]] = [[(-9*1)+(-11*2), (-9*-1)+(-11*3)], [(22*1)+(13*2), (22*-1)+(13*3)]] = [[-9-22, 9-33], [22+26, -22+39]] = [[-31, -24], [48, 17]].

Untuk b. Menentukan hasil P² - P³ + P⁴:
   P² - P³ + P⁴ = [[-1, -4], [8, 7]] - [[-9, -11], [22, 13]] + [[-31, -24], [48, 17]]
   = [[(-1 - (-9) - 31), (-4 - (-11) - 24)], [(8 - 22 + 48), (7 - 13 + 17)]]
   = [[(-1 + 9 - 31), (-4 + 11 - 24)], [(8 - 22 + 48), (7 - 13 + 17)]]
   = [[(8 - 31), (7 - 24)], [(-14 + 48), (-6 + 17)]]
   = [[-23, -17], [34, 11]].