Diketahui matriks-matriks P=(2 5 1 3) dan Q=(5 4 1 1). Jika

P^(-1)Q^(-1) = [[8, -37], [-3, 14]], Det(P^(-1)Q^(-1)) = 1

Pembahasan

Diketahui matriks P = [[2, 5], [1, 3]] dan Q = [[5, 4], [1, 1]].

Pertama, kita cari invers dari matriks P (P^-1):
Determinan P (det(P)) = (2 * 3) - (5 * 1) = 6 - 5 = 1
P^-1 = 1/det(P) * [[3, -5], [-1, 2]] = 1/1 * [[3, -5], [-1, 2]] = [[3, -5], [-1, 2]]

Selanjutnya, kita cari invers dari matriks Q (Q^-1):
Determinan Q (det(Q)) = (5 * 1) - (4 * 1) = 5 - 4 = 1
Q^-1 = 1/det(Q) * [[1, -4], [-1, 5]] = 1/1 * [[1, -4], [-1, 5]] = [[1, -4], [-1, 5]]

Sekarang, kita cari hasil perkalian P^-1 * Q^-1:
P^-1 * Q^-1 = [[3, -5], [-1, 2]] * [[1, -4], [-1, 5]]
= [[(3*1 + -5*-1), (3*-4 + -5*5)], [(-1*1 + 2*-1), (-1*-4 + 2*5)]]
= [[(3 + 5), (-12 - 25)], [(-1 - 2), (4 + 10)]]
= [[8, -37], [-3, 14]]

Terakhir, kita cari determinan dari matriks P^-1 * Q^-1:
Det(P^-1 * Q^-1) = (8 * 14) - (-37 * -3)
= 112 - 111
= 1

Jadi, matriks P^-1 Q^-1 adalah [[8, -37], [-3, 14]] dan determinan matriks P^-1 Q^-1 adalah 1.