Tentukan inversi dari matriks A=[2 4 -2 -1 2 1 3 -3 4]!

A^-1 = [[11/56, -5/28, 1/7], [1/8, 1/4, 0], [-3/56, 9/28, 1/7]]

Pembahasan

Untuk mencari inversi dari matriks A=[[2, 4, -2], [-1, 2, 1], [3, -3, 4]], kita perlu menggunakan rumus invers matriks 3x3. Pertama, kita hitung determinan (det(A)).
det(A) = 2 * ((2*4) - (1*-3)) - 4 * ((-1*4) - (1*3)) + (-2) * ((-1*-3) - (2*3))
det(A) = 2 * (8 - (-3)) - 4 * (-4 - 3) - 2 * (3 - 6)
det(A) = 2 * (11) - 4 * (-7) - 2 * (-3)
det(A) = 22 + 28 + 6
det(A) = 56

Karena determinan tidak nol (det(A) = 56 ≠ 0), maka matriks A memiliki invers.
Selanjutnya, kita perlu mencari matriks adjoin (adj(A)). Matriks adjoin adalah transpose dari matriks kofaktor.

Menghitung matriks kofaktor (C):
C11 = +( (2*4) - (1*-3) ) = 8 + 3 = 11
C12 = -( (-1*4) - (1*3) ) = -(-4 - 3) = -(-7) = 7
C13 = +( (-1*-3) - (2*3) ) = 3 - 6 = -3
C21 = -( (4*4) - (-2*-3) ) = -(16 - 6) = -10
C22 = +( (2*4) - (-2*3) ) = 8 - (-6) = 8 + 6 = 14
C23 = -( (2*-3) - (4*3) ) = -(-6 - 12) = -(-18) = 18
C31 = +( (4*1) - (-2*2) ) = 4 - (-4) = 4 + 4 = 8
C32 = -( (2*1) - (-2*-1) ) = -(2 - 2) = 0
C33 = +( (2*2) - (4*-1) ) = 4 - (-4) = 4 + 4 = 8

Matriks Kofaktor (C) = [[11, 7, -3], [-10, 14, 18], [8, 0, 8]]

Matriks Adjoin (adj(A)) adalah transpose dari C:
adj(A) = C^T = [[11, -10, 8], [7, 14, 0], [-3, 18, 8]]

Invers matriks A (A^-1) adalah (1/det(A)) * adj(A):
A^-1 = (1/56) * [[11, -10, 8], [7, 14, 0], [-3, 18, 8]]

A^-1 = [[11/56, -10/56, 8/56], [7/56, 14/56, 0/56], [-3/56, 18/56, 8/56]]

Sederhanakan pecahan jika memungkinkan:
A^-1 = [[11/56, -5/28, 1/7], [1/8, 1/4, 0], [-3/56, 9/28, 1/7]]