The document discusses the calculation of the inverse of a 3x3 matrix. It defines the minor, cofactor, adjoint and determinant of a matrix which are used to calculate the inverse. The inverse of a 3x3 matrix A is calculated as the adjoint of A divided by the determinant of A. An example calculates each component and shows the full inverse of a 3x3 matrix.

1. INVERSMATRIKSPERSERGI BERORDO 3
A. MINOR
Minor dituliskanMij besarnya | 𝑀𝑖𝑗| dimanai = banyakbaris + J = banyakkolom
Contohtentukanminorminordari matriks
𝐴 = [
6 −1 1
−2 −3 1
3 2 −2
]
𝑚𝑖𝑛𝑜𝑟 𝑎11 → 𝑀11 = |
−3 1
2 −1
| = 3 − 2 = 1
𝑚𝑖𝑛𝑜𝑟 𝑎12 → 𝑀12 = |
−2 1
3 −1
| = 2 − 3 = −1
𝑚𝑖𝑛𝑜𝑟 𝑎13 → 𝑀13 = |
−2 −3
3 2
| = −4 + 9 = 5
𝐴 = [
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
]
𝑚𝑖𝑛𝑜𝑟 𝑎21 → 𝑀21 = |
−1 1
2 −1
| = 1 − 2 = −1
𝑚𝑖𝑛𝑜𝑟 𝑎22 → 𝑀22 = |
6 1
3 −1
| = −6 − 3 = −9
𝑚𝑖𝑛𝑜𝑟 𝑎23 → 𝑀23 = |
6 −1
3 2
| = 12 + 3 = 15
𝑚𝑖𝑛𝑜𝑟 𝑎31 → 𝑀31 = |
−1 1
−3 1
| = −1 + 3 = 2
𝑚𝑖𝑛𝑜𝑟 𝑎32 → 𝑀32 = |
6 1
−2 1
| = 6 + 2 = 8
𝑚𝑖𝑛𝑜𝑟 𝑎33 → 𝑀33 = |
6 −1
−2 −3
| = −18 − 2 = −20
B. KOFAKTOR
Kofaktordituliskanαij besarnya∝ 𝑖𝑗= (−1) 𝑖+𝑗.| 𝑀𝑖𝑗|
I = banyaknyabarisdanj = banyakkolom
Contohtentukankofaktorkofaktordari matriks
𝐴 = [
6 −1 1
−2 −3 1
3 2 −2
]
Kofaktor ∝11→ (−1)1+1.| 𝑀11| = (−1)2.1 = 1
∝12→ (−1)1+2.| 𝑀12| = (−1)3.−1 = 1
∝13→ (−1)1+3.| 𝑀13| = (−1)4.5 = 5
∝21→ (−1)2+1.| 𝑀21| = (−1)3.−1 = 1
∝22→ (−1)2+2.| 𝑀22| = (−1)4.−9 = −9
∝23→ (−1)2+3.| 𝑀23| = (−1)5.15 = −15
∝31→ (−1)3+1.| 𝑀31| = (−1)4.2 = 2
∝32→ (−1)3+2.| 𝑀32| = (−1)5.8 = −8
∝33→ (−1)3+3.| 𝑀33| = (−1)6.−20 = −20
C. ADJOINT

2. 𝑎𝑑𝑗 ( 𝐴) = [
∝11 ∝12 ∝13
∝21 ∝22 ∝23
∝31 ∝32 ∝33
]
ContohtentukanAdjointdari matriks
𝐴 = [
6 −1 1
−2 −3 1
3 2 −2
]
𝑎𝑑𝑗 ( 𝐴) = [
∝11 ∝21 ∝31
∝12 ∝22 ∝32
∝13 ∝23 ∝33
] = [
1 1 2
1 −9 −8
5 −15 −20
]
D. DeterminanMatriksordo3x3
Penjabaranmenjadi determinanordo2x2
det( 𝐴) = | 𝐴| = 𝑎11 |
𝑎22 𝑎23
𝑎32 𝑎33
|− 𝑎12 |
𝑎21 𝑎23
𝑎31 𝑎33
|+ 𝑎13 |
𝑎21 𝑎22
𝑎31 𝑎32
|
Contohhitunglahnilai determinandari matriks
𝐴 = [
6 −1 1
−2 −3 1
3 2 −2
]
det( 𝐴) = 6 |
−3 1
2 −1
| − (−1) |
−2 1
3 −1
| + 1|
−2 −3
3 2
|
= 6(3 − 2) + 1(2 − 3) + 1(−4 + 9)
= 6(1) + 1(−1) + 1(5)
= 10
E. InversMatriksOrdo 3x3
InversMatriksA ordo 3x3 ditentukan
𝐴−1 =
1
det( 𝐴)
. 𝑎𝑑𝑗 ( 𝐴)
Contohtentukaninvers matriksdari matriksberikut
𝐴 = [
6 −1 1
−2 −3 1
3 2 −2
]
det( 𝐴) = 10
𝑎𝑑𝑗 ( 𝐴) = [
1 1 2
1 −9 −8
5 −15 −20
]
𝐴−1 =
1
10
. [
1 1 2
1 −9 −8
5 −15 −20
] =
[
1
10
1
10
2
10
1
10
−9
10
−8
10
5
10
−15
10
−20
10 ]
=
[
1
10
1
10
1
5
1
10
−9
10
−4
5
1
2
−3
2
−2]