Diketahui matriks-matriks A(3 5 -2 -3) B=(3 4 5 7) a.

a. AB=[[34, 47], [-21, -29]], BA=[[1, 3], [1, 4]], (AB)^-1=[[-29, -47], [21, 34]], (BA)^-1=[[4, -3], [-1, 1]]. b. A^-1=[[-3, -5], [2, 3]], B^-1=[[7, -4], [-5, 3]], A^-1B^-1=[[4, -3], [-1, 1]], B^-1A^-1=[[-29, -47], [21, 34]]. c. Ya, kedua pernyataan terbukti benar.

Pembahasan

Diberikan matriks A = [[3, 5], [-2, -3]] dan B = [[3, 4], [5, 7]].

a. Menentukan AB, BA, (AB)^-1, dan (BA)^-1:

AB = [[3, 5], [-2, -3]] * [[3, 4], [5, 7]]
   = [[(3*3)+(5*5), (3*4)+(5*7)], [(-2*3)+(-3*5), (-2*4)+(-3*7)]]
   = [[9+25, 12+35], [-6-15, -8-21]]
   = [[34, 47], [-21, -29]]

Determinan AB (det(AB)) = (34 * -29) - (47 * -21)
                          = -986 - (-987)
                          = -986 + 987
                          = 1

(AB)^-1 = 1/det(AB) * [[-29, -47], [21, 34]]
        = 1/1 * [[-29, -47], [21, 34]]
        = [[-29, -47], [21, 34]]

BA = [[3, 4], [5, 7]] * [[3, 5], [-2, -3]]
   = [[(3*3)+(4*-2), (3*5)+(4*-3)], [(5*3)+(7*-2), (5*5)+(7*-3)]]
   = [[9-8, 15-12], [15-14, 25-21]]
   = [[1, 3], [1, 4]]

Determinan BA (det(BA)) = (1 * 4) - (3 * 1)
                          = 4 - 3
                          = 1

(BA)^-1 = 1/det(BA) * [[4, -3], [-1, 1]]
        = 1/1 * [[4, -3], [-1, 1]]
        = [[4, -3], [-1, 1]]

b. Menentukan A^-1, B^-1, A^-1B^-1, dan B^-1A^-1:

Determinan A (det(A)) = (3 * -3) - (5 * -2)
                       = -9 - (-10)
                       = -9 + 10
                       = 1

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

Determinan B (det(B)) = (3 * 7) - (4 * 5)
                       = 21 - 20
                       = 1

B^-1 = 1/det(B) * [[7, -4], [-5, 3]]
     = 1/1 * [[7, -4], [-5, 3]]
     = [[7, -4], [-5, 3]]

A^-1B^-1 = [[-3, -5], [2, 3]] * [[7, -4], [-5, 3]]
          = [[(-3*7)+(-5*-5), (-3*-4)+(-5*3)], [(2*7)+(3*-5), (2*-4)+(3*3)]]
          = [[-21+25, 12-15], [14-15, -8+9]]
          = [[4, -3], [-1, 1]]

B^-1A^-1 = [[7, -4], [-5, 3]] * [[-3, -5], [2, 3]]
          = [[(7*-3)+(-4*2), (7*-5)+(-4*3)], [(-5*-3)+(3*2), (-5*-5)+(3*3)]]
          = [[-21-8, -35-12], [15+6, 25+9]]
          = [[-29, -47], [21, 34]]

c. Memeriksa hubungan:

(i) Apakah (AB)^-1 = B^-1A^-1?
   (AB)^-1 = [[-29, -47], [21, 34]]
   B^-1A^-1 = [[-29, -47], [21, 34]]
   Ya, (AB)^-1 = B^-1A^-1.

(ii) Apakah (BA)^-1 = A^-1B^-1?
    (BA)^-1 = [[4, -3], [-1, 1]]
    A^-1B^-1 = [[4, -3], [-1, 1]]
    Ya, (BA)^-1 = A^-1B^-1.

Hasil perhitungan mengkonfirmasi bahwa sifat-sifat invers matriks tersebut berlaku.