Dari Wikibuku bahasa Indonesia, sumber buku teks bebas
bentuk
ordo 2x2:
[
a
b
c
d
]
{\displaystyle {\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}}
ordo 3x3:
[
a
b
c
d
e
f
g
h
i
]
{\displaystyle {\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}}
sifat
komutatif
asosiatif
(A + B) + C = A + (B + C)
(A . B) . C = A. (B x C)
distributif
A . (B + C) = A . B + A . C
A . (B - C) = A . B - A . C
(k . A) . B = k. (A . B)
A . B ≠ B . A
A . I = A
A . A-1 = I
(A . B)-1 = B-1 . A-1
(A . B . C)-1 = C-1 . B-1 . A-1
det (A-1 ) = det A (invers bukan pangkat)
det (AT ) = 1/det A
det (An ) = (det A)n
vektor baris
[
3
7
2
]
{\displaystyle {\begin{bmatrix}3&7&2\end{bmatrix}}}
vektor kolom
[
4
1
8
]
{\displaystyle {\begin{bmatrix}4\\1\\8\\\end{bmatrix}}}
matriks persegi
[
9
13
5
1
11
7
2
6
3
]
{\displaystyle {\begin{bmatrix}9&13&5\\1&11&7\\2&6&3\\\end{bmatrix}}}
[
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
]
{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}}
pertama: a11, a12 dan a13
kedua: a21, a22 dan a23
ketiga: a31, a32 dan a33
pertama: a11, a21 dan a31
kedua: a12, a22 dan a32
ketiga: a13, a23 dan a33
sisi kiri ke kanan: a11, a22 dan a33
sisi kanan ke kiri: a13, a22 dan a31
[
a
11
0
0
0
a
22
0
0
0
a
33
]
{\displaystyle {\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\\\end{bmatrix}}}
[
a
11
0
0
a
21
a
22
0
a
31
a
32
a
33
]
{\displaystyle {\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}}
[
a
11
a
12
a
13
0
a
22
a
23
0
0
a
33
]
{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\\\end{bmatrix}}}
[
a
b
c
d
]
⋅
[
e
f
g
h
]
{\displaystyle {\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}\cdot {\begin{bmatrix}e&f\\g&h\\\end{bmatrix}}}
=
[
a
e
+
b
g
a
f
+
b
h
c
e
+
d
g
c
f
+
d
h
]
{\displaystyle {\begin{bmatrix}ae+bg&af+bh\\ce+dg&cf+dh\\\end{bmatrix}}}
bentuk
[
a
b
c
d
]
{\displaystyle {\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}}
Matriks transpos (AT )
[
a
c
b
d
]
{\displaystyle {\begin{bmatrix}a&c\\b&d\end{bmatrix}}}
Determinan (Det)
ad - bc
Matriks singular adalah matriks yang hasil determinan bernilai nol sedangkan matriks nonsingular adalah matriks yang hasil determinan bernilai bukan nol.
Matriks inverse (A-1 )
1
det A
{\displaystyle {\frac {1}{\text{det A}}}}
[
d
−
b
−
c
a
]
{\displaystyle {\begin{bmatrix}d&-b\\-c&a\\\end{bmatrix}}}
bentuk
[
a
b
c
d
e
f
g
h
i
]
{\displaystyle {\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}}
Matriks transpos (AT )
[
a
d
g
b
e
h
c
f
i
]
{\displaystyle {\begin{bmatrix}a&d&g\\b&e&h\\c&f&i\\\end{bmatrix}}}
Determinan (Det)
a
b
c
a
b
d
e
f
d
e
g
h
i
g
h
−
−
−
+
+
+
{\displaystyle {\begin{aligned}{\begin{array}{rrr|rr}a&b&c&a&b\\d&e&f&d&e\\g&h&i&g&h\\-&-&-+&+&+\\\end{array}}\\\end{aligned}}}
det A = aei + bfg + cdh - bdi - afh - ceg
untuk minor Mij = det Aij
untuk kofaktor Cij = (-1)i+j . Mij
det A =
∑
j
=
1
n
i
=
a
i
j
⋅
C
i
j
{\displaystyle \sum _{j=1}^{n}i=a_{ij}\cdot C_{ij}}
dimana sembarang baris i atau kolom j (i atau j = 1, 2, 3, ..., n)
Adjoint (Adj)
kof (A) = (-1)i+j . Mij
kof (A) =
[
M
11
M
12
M
13
M
21
M
22
M
23
M
31
M
32
M
33
]
{\displaystyle {\begin{bmatrix}M_{11}&M_{12}&M_{13}\\M_{21}&M_{22}&M_{23}\\M_{31}&M_{32}&M_{33}\\\end{bmatrix}}}
adj A = (kof (A))T
Matriks inverse (A-1 )
a
d
j
A
det A
{\displaystyle {\frac {adjA}{\text{det A}}}}
A | I diubah menjadi I | A-1
contoh
tentukan hasil determinan serta matriks invers dari
[
2
1
6
4
]
{\displaystyle {\begin{bmatrix}2&1\\6&4\\\end{bmatrix}}}
!
Jawaban
∗
d
e
t
A
=
2
⋅
4
−
1
⋅
6
=
8
−
6
=
2
∗
A
−
1
=
1
d
e
t
A
⋅
[
4
−
1
−
6
2
]
=
1
2
⋅
[
4
−
1
−
6
2
]
=
[
2
−
1
2
−
3
1
]
{\displaystyle {\begin{aligned}*detA&=2\cdot 4-1\cdot 6=8-6=2\\*A^{-1}&={\frac {1}{detA}}\cdot {\begin{bmatrix}4&-1\\-6&2\\\end{bmatrix}}\\&={\frac {1}{2}}\cdot {\begin{bmatrix}4&-1\\-6&2\\\end{bmatrix}}\\&={\begin{bmatrix}2&-{\frac {1}{2}}\\-3&1\\\end{bmatrix}}\\\end{aligned}}}
tentukan hasil determinan serta matriks invers dari
[
1
2
3
2
8
7
1
5
6
]
{\displaystyle {\begin{bmatrix}1&2&3\\2&8&7\\1&5&6\\\end{bmatrix}}}
!
Jawaban
∗
cara 1
1
2
3
1
2
2
8
7
2
8
1
5
6
1
5
−
−
−
+
+
+
d
e
t
A
=
1
⋅
8
⋅
6
+
2
⋅
7
⋅
1
+
3
⋅
2
⋅
5
−
2
⋅
2
⋅
6
−
1
⋅
7
⋅
5
−
3
⋅
8
⋅
1
=
48
+
14
+
30
−
24
−
35
−
24
=
9
cara 2
d
e
t
A
=
a
11
⋅
c
11
+
a
12
⋅
c
12
+
a
13
⋅
c
13
=
1
⋅
(
−
1
)
1
+
1
⋅
[
8
7
5
6
]
+
2
⋅
(
−
1
)
1
+
2
⋅
[
2
7
1
6
]
+
3
⋅
(
−
1
)
1
+
3
⋅
[
2
8
1
5
]
=
1
⋅
1
⋅
(
48
−
35
)
+
2
⋅
(
−
1
)
⋅
(
12
−
7
)
+
3
⋅
1
⋅
(
10
−
8
)
=
13
−
10
+
6
=
9
∗
cara 1
k
o
f
(
A
)
=
[
[
8
7
5
6
]
−
[
2
7
1
6
]
[
2
8
1
5
]
−
[
2
3
5
6
]
[
1
3
1
6
]
−
[
1
2
1
5
]
[
2
3
8
7
]
−
[
1
3
2
7
]
[
1
2
2
8
]
]
=
[
13
−
5
2
3
3
−
3
−
10
−
1
4
]
a
d
j
(
A
)
=
(
k
o
f
(
A
)
)
T
=
[
13
3
−
10
−
5
3
−
1
2
−
3
4
]
A
−
1
=
a
d
j
A
d
e
t
A
=
1
9
⋅
[
13
3
−
10
−
5
3
−
1
2
−
3
4
]
=
[
13
9
1
3
−
10
9
−
5
9
1
3
−
1
9
2
9
−
1
3
4
9
]
cara 2
1
2
3
1
0
0
2
8
7
0
1
0
1
5
6
0
0
1
b2-2b1
1
2
3
1
0
0
0
4
1
−
2
1
0
1
5
6
0
0
1
b3-b1
1
2
3
1
0
0
0
4
1
−
2
1
0
0
3
3
−
1
0
1
1/4b2
1
2
3
1
0
0
0
1
1
4
−
1
2
1
4
0
0
3
3
−
1
0
1
b1-2b2
1
0
5
2
2
−
1
2
0
0
1
1
4
−
1
2
1
4
0
0
3
3
−
1
0
1
b3-3b2
1
0
5
2
2
−
1
2
0
0
1
1
4
−
1
2
1
4
0
0
0
9
4
1
2
−
3
4
1
4/9b3
1
0
5
2
2
−
1
2
0
0
1
1
4
−
1
2
1
4
0
0
0
1
2
9
−
1
3
4
9
b1-5/2b3
1
0
0
13
9
1
3
−
10
9
0
1
1
4
−
1
2
1
4
0
0
0
1
2
9
−
1
3
4
9
b2-1/4b3
1
0
0
13
9
1
3
−
10
9
0
1
0
−
5
9
1
3
−
1
9
0
0
1
2
9
−
1
3
4
9
jadi
A
−
1
=
[
13
9
1
3
−
10
9
−
5
9
1
3
−
1
9
2
9
−
1
3
4
9
]
{\displaystyle {\begin{aligned}*{\text{cara 1}}\\{\begin{array}{rrr|rr}1&2&3&1&2\\2&8&7&2&8\\1&5&6&1&5\\-&-&-+&+&+\\\end{array}}\\detA&=1\cdot 8\cdot 6+2\cdot 7\cdot 1+3\cdot 2\cdot 5-2\cdot 2\cdot 6-1\cdot 7\cdot 5-3\cdot 8\cdot 1=48+14+30-24-35-24=9\\{\text{cara 2}}\\detA&=a_{11}\cdot c_{11}+a_{12}\cdot c_{12}+a_{13}\cdot c_{13}\\&=1\cdot (-1)^{1+1}\cdot {\begin{bmatrix}8&7\\5&6\\\end{bmatrix}}+2\cdot (-1)^{1+2}\cdot {\begin{bmatrix}2&7\\1&6\\\end{bmatrix}}+3\cdot (-1)^{1+3}\cdot {\begin{bmatrix}2&8\\1&5\\\end{bmatrix}}\\&=1\cdot 1\cdot (48-35)+2\cdot (-1)\cdot (12-7)+3\cdot 1\cdot (10-8)\\&=13-10+6=9\\*{\text{cara 1}}\\kof(A)&={\begin{bmatrix}{\begin{bmatrix}8&7\\5&6\\\end{bmatrix}}&-{\begin{bmatrix}2&7\\1&6\\\end{bmatrix}}&{\begin{bmatrix}2&8\\1&5\\\end{bmatrix}}\\-{\begin{bmatrix}2&3\\5&6\\\end{bmatrix}}&{\begin{bmatrix}1&3\\1&6\\\end{bmatrix}}&-{\begin{bmatrix}1&2\\1&5\\\end{bmatrix}}\\{\begin{bmatrix}2&3\\8&7\\\end{bmatrix}}&-{\begin{bmatrix}1&3\\2&7\\\end{bmatrix}}&{\begin{bmatrix}1&2\\2&8\\\end{bmatrix}}\\\end{bmatrix}}\\&={\begin{bmatrix}13&-5&2\\3&3&-3\\-10&-1&4\\\end{bmatrix}}\\adj(A)&=(kof(A))^{T}\\&={\begin{bmatrix}13&3&-10\\-5&3&-1\\2&-3&4\\\end{bmatrix}}\\A^{-1}&={\frac {adjA}{detA}}\\&={\frac {1}{9}}\cdot {\begin{bmatrix}13&3&-10\\-5&3&-1\\2&-3&4\\\end{bmatrix}}\\&={\begin{bmatrix}{\frac {13}{9}}&{\frac {1}{3}}&-{\frac {10}{9}}\\-{\frac {5}{9}}&{\frac {1}{3}}&-{\frac {1}{9}}\\{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{bmatrix}}\\{\text{cara 2}}\\{\begin{array}{rrr|rrr}1&2&3&1&0&0\\2&8&7&0&1&0\\1&5&6&0&0&1\\\end{array}}\\{\text{b2-2b1}}\\{\begin{array}{rrr|rrr}1&2&3&1&0&0\\0&4&1&-2&1&0\\1&5&6&0&0&1\\\end{array}}\\{\text{b3-b1}}\\{\begin{array}{rrr|rrr}1&2&3&1&0&0\\0&4&1&-2&1&0\\0&3&3&-1&0&1\\\end{array}}\\{\text{1/4b2}}\\{\begin{array}{rrr|rrr}1&2&3&1&0&0\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&3&3&-1&0&1\\\end{array}}\\{\text{b1-2b2}}\\{\begin{array}{rrr|rrr}1&0&{\frac {5}{2}}&2&-{\frac {1}{2}}&0\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&3&3&-1&0&1\\\end{array}}\\{\text{b3-3b2}}\\{\begin{array}{rrr|rrr}1&0&{\frac {5}{2}}&2&-{\frac {1}{2}}&0\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&0&{\frac {9}{4}}&{\frac {1}{2}}&-{\frac {3}{4}}&1\\\end{array}}\\{\text{4/9b3}}\\{\begin{array}{rrr|rrr}1&0&{\frac {5}{2}}&2&-{\frac {1}{2}}&0\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&0&1&{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{array}}\\{\text{b1-5/2b3}}\\{\begin{array}{rrr|rrr}1&0&0&{\frac {13}{9}}&{\frac {1}{3}}&-{\frac {10}{9}}\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&0&1&{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{array}}\\{\text{b2-1/4b3}}\\{\begin{array}{rrr|rrr}1&0&0&{\frac {13}{9}}&{\frac {1}{3}}&-{\frac {10}{9}}\\0&1&0&-{\frac {5}{9}}&{\frac {1}{3}}&-{\frac {1}{9}}\\0&0&1&{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{array}}\\{\text{jadi}}A^{-1}&={\begin{bmatrix}{\frac {13}{9}}&{\frac {1}{3}}&-{\frac {10}{9}}\\-{\frac {5}{9}}&{\frac {1}{3}}&-{\frac {1}{9}}\\{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{bmatrix}}\\\end{aligned}}}