Diketahui persamaan matriks AX=B dan BC=D. Jika A = (2 5 1

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Pembahasan

Diketahui persamaan matriks AX = B dan BC = D. Kita juga diberikan matriks A = [[2, 5], [1, 3]], C = [[1, 1], [1, 0]], dan D = [[2, 4], [1, 3]].

Langkah 1: Cari matriks B dari persamaan BC = D.
Kita perlu mencari invers dari C, yaitu C⁻¹.
Determinan C (det(C)) = (1 * 0) - (1 * 1) = -1.

C⁻¹ = (1 / det(C)) * [[0, -1], [-1, 1]] = (1 / -1) * [[0, -1], [-1, 1]] = [[0, 1], [1, -1]].

Sekarang, B = D * C⁻¹.
B = [[2, 4], [1, 3]] * [[0, 1], [1, -1]]
B = [[(2*0 + 4*1), (2*1 + 4*(-1))], [(1*0 + 3*1), (1*1 + 3*(-1))]]
B = [[4, -2], [3, -2]].

Langkah 2: Cari matriks X dari persamaan AX = B.
Kita perlu mencari invers dari A, yaitu A⁻¹.
Determinan A (det(A)) = (2 * 3) - (5 * 1) = 6 - 5 = 1.

A⁻¹ = (1 / det(A)) * [[3, -5], [-1, 2]] = (1 / 1) * [[3, -5], [-1, 2]] = [[3, -5], [-1, 2]].

Sekarang, X = A⁻¹ * B.
X = [[3, -5], [-1, 2]] * [[4, -2], [3, -2]]
X = [[(3*4 + (-5)*3), (3*(-2) + (-5)*(-2))], [((-1)*4 + 2*3), ((-1)*(-2) + 2*(-2))]]
X = [[(12 - 15), (-6 + 10)], [(-4 + 6), (2 - 4)]]
X = [[-3, 4], [2, -2]].

Langkah 3: Hitung determinan X.
Determinan X (det(X)) = (-3 * -2) - (4 * 2) = 6 - 8 = -2.