Diketahui A=(3 -5 4 -7) dan B=(-9 5 -7 4). Tentukan a.

a. (AB)^-1 = [-69/38 25/19; -71/38 26/19]. b. (BA)^-1 = [-12/19 5/19; -13/38 7/38].

Pembahasan

Diberikan matriks A = (3 -5 4 -7) dan B = (-9 5 -7 4).
Kita perlu menentukan (AB)^-1 dan (BA)^-1.

Langkah 1: Hitung perkalian matriks AB.
AB = 
[ 3  -5 ] [ -9   5 ]
[ 4  -7 ] [  5  -7 ]

Elemen AB(1,1) = (3)(-9) + (-5)(5) = -27 - 25 = -52
Elemen AB(1,2) = (3)(5) + (-5)(-7) = 15 + 35 = 50
Elemen AB(2,1) = (4)(-9) + (-7)(5) = -36 - 35 = -71
Elemen AB(2,2) = (4)(5) + (-7)(-7) = 20 + 49 = 69

Jadi, AB = 
[ -52   50 ]
[ -71   69 ]

Langkah 2: Hitung invers dari AB, yaitu (AB)^-1.
Untuk matriks 2x2 [ a b ; c d ], inversnya adalah (1/det) * [ d -b ; -c a ], di mana det = ad - bc.

determinant(AB) = (-52)(69) - (50)(-71)
= -3588 - (-3550)
= -3588 + 3550
= -38

(AB)^-1 = (1 / -38) * 
[  69  -50 ]
[  71  -52 ]

(AB)^-1 = 
[  69/-38   -50/-38 ]
[  71/-38   -52/-38 ]

(AB)^-1 = 
[ -69/38   50/38 ]
[ -71/38   52/38 ]

(AB)^-1 = 
[ -69/38   25/19 ]
[ -71/38   26/19 ]

Langkah 3: Hitung perkalian matriks BA.
BA = 
[ -9   5 ] [ 3  -5 ]
[  5  -7 ] [ 4  -7 ]

Elemen BA(1,1) = (-9)(3) + (5)(4) = -27 + 20 = -7
Elemen BA(1,2) = (-9)(-5) + (5)(-7) = 45 - 35 = 10
Elemen BA(2,1) = (5)(3) + (-7)(4) = 15 - 28 = -13
Elemen BA(2,2) = (5)(-5) + (-7)(-7) = -25 + 49 = 24

Jadi, BA = 
[ -7   10 ]
[ -13  24 ]

Langkah 4: Hitung invers dari BA, yaitu (BA)^-1.

determinant(BA) = (-7)(24) - (10)(-13)
= -168 - (-130)
= -168 + 130
= -38

(BA)^-1 = (1 / -38) * 
[  24  -10 ]
[  13  -7 ]

(BA)^-1 = 
[  24/-38   -10/-38 ]
[  13/-38   -7/-38 ]

(BA)^-1 = 
[ -24/38   10/38 ]
[ -13/38    7/38 ]

(BA)^-1 = 
[ -12/19   5/19 ]
[ -13/38    7/38 ]