Jika A=(1 3 3 4) dan B=(2 2 1 3), maka (AB)^-1 A^T= .....

[[3/4, -1/2], [-1/4, 1/2]]

Pembahasan

Untuk menghitung (AB)^-1 A^T, kita perlu melakukan beberapa langkah:

Diketahui matriks A = [[1, 3], [3, 4]] dan B = [[2, 2], [1, 3]].

Langkah 1: Hitung hasil perkalian matriks AB.
AB = [[1*2 + 3*1, 1*2 + 3*3], [3*2 + 4*1, 3*2 + 4*3]]
AB = [[2 + 3, 2 + 9], [6 + 4, 6 + 12]]
AB = [[5, 11], [10, 18]]

Langkah 2: Hitung invers dari matriks AB, yaitu (AB)^-1.
Determinan dari AB = (5 * 18) - (11 * 10) = 90 - 110 = -20.
(AB)^-1 = 1/(-20) * [[18, -11], [-10, 5]]
(AB)^-1 = [[-18/20, 11/20], [10/20, -5/20]]
(AB)^-1 = [[-9/10, 11/20], [1/2, -1/4]]

Langkah 3: Tentukan transpose dari matriks A, yaitu A^T.
A^T = [[1, 3], [3, 4]]

Langkah 4: Kalikan (AB)^-1 dengan A^T.
(AB)^-1 A^T = [[-9/10, 11/20], [1/2, -1/4]] * [[1, 3], [3, 4]]

Elemen baris 1, kolom 1:
(-9/10 * 1) + (11/20 * 3) = -9/10 + 33/20 = -18/20 + 33/20 = 15/20 = 3/4

Elemen baris 1, kolom 2:
(-9/10 * 3) + (11/20 * 4) = -27/10 + 44/20 = -54/20 + 44/20 = -10/20 = -1/2

Elemen baris 2, kolom 1:
(1/2 * 1) + (-1/4 * 3) = 1/2 - 3/4 = 2/4 - 3/4 = -1/4

Elemen baris 2, kolom 2:
(1/2 * 3) + (-1/4 * 4) = 3/2 - 4/4 = 3/2 - 1 = 1/2

Hasilnya adalah: [[3/4, -1/2], [-1/4, 1/2]]