materi matriks smk ap
DESCRIPTION
hehe kali ini saya akan beri kalian file lagi nih tentang matematika hehe lebih tepatnya ke materi matriks. disini diulas bagaimana ciri ciri matriks dan rumus rumus tentang matriks. file ini berbentuk ppt jadiTRANSCRIPT
![Page 1: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/1.jpg)
1
MATRIKS
![Page 2: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/2.jpg)
Matrix asalah susunan bilangan berbentuk segi-4 yang terdiri atas baris dan kolom yang ditulis dalam sepasang tanda kurung.
DEFINISI
![Page 3: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/3.jpg)
NOTASI OF MATRIKS
mnmm
n
n
m aaa
aaaaaa
a
aa
A
...............
...
...
...
32
22322
11312
1
21
11
Nama Matriks
Amxn
![Page 4: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/4.jpg)
ELEMENT MATRIKS
mnmm
n
n
m aaa
aaaaaa
a
aa
A
...............
...
...
...
32
22322
11312
1
21
11elementbaris
1
Letak elemenElemen kolom 1
![Page 5: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/5.jpg)
ORDO
mnmm
n
n
m aaa
aaaaaa
a
aa
A
...............
...
...
...
32
22322
11312
1
21
11 Baris 1
Baris 2
Baris m
Kolom1 Kolom2
Kolom 3 Kolom n
Ordo m x n Notasi : A m x n
![Page 6: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/6.jpg)
53
12
643970182
Ζ
1. Apakah nama matriks di atas?
Contoh:
![Page 7: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/7.jpg)
53
12
643970182
Ζ
2. Sebutkan elemen baris 3 dan kolom 4!
![Page 8: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/8.jpg)
53
12
643970182
Ζ
3. Sebutkan elemen baris ke-2!
![Page 9: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/9.jpg)
53
12
643970182
Ζ
3. Sebutkan ordo matriks di atas dan notasinya!
Ordo 3 x 4
Notasi : Z 3 x 4
![Page 10: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/10.jpg)
JENIS-JENIS MATRIKS
10
![Page 11: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/11.jpg)
MATRIKS BARIS
4991N
![Page 12: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/12.jpg)
MATRIKS KOLOM
603
S
![Page 13: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/13.jpg)
MATRIKS DIAGONAL
100010004
M
![Page 14: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/14.jpg)
MATRIKS IDENTITAS
000
000
W
100010001
W
Penjumlahan Perkalian
Matriks 0
1001
W
![Page 15: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/15.jpg)
MATRIKS SEGITIGA
14002110341
W
Segitiga Atas
Segitiga bawah
28472030600290001
W
![Page 16: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/16.jpg)
TRANSPOS MATRIKS
Transpos matriks A terjadi jika setiap baris pada matriks tersebut berubah menjadi kolom . Transpose matriks A ditulis A’ atau At. Sehingga A m x n menjadi A’ n x m.
Elemen baris 1 matriks A Kolom 1 matriks A’
Elemen baris 2 matriks A Kolom 2 matriks A’
dst
![Page 17: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/17.jpg)
TRANSPOS MATRIKS
93
214
105
21'A
A 4 x 2
921102
3451
A
A’ 2 x
4
93
214
105
21tA
![Page 18: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/18.jpg)
Tentukan transpose matriks berikut!
125
A
413221130
B
![Page 19: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/19.jpg)
Tentukan transpose matriks berikut!
125
tA 125 A
![Page 20: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/20.jpg)
Jawab
421123310
B
413221130
tB
413221130
B
![Page 21: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/21.jpg)
hgc
fe
Bdb
aA
a = e
d = h
b = fc = g
![Page 22: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/22.jpg)
PERSAMAAN MATRIKS
zx
2-1
B32-
1A
4
If A = B, tentukan nilai x dan z!
![Page 23: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/23.jpg)
y4x2x52
Bz6yx2
A
Jika A = B, tentukan nilai x, y dan z!
![Page 24: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/24.jpg)
y4x2x52
Bz6yx2
A
2 = 2
z = 4x - y
x + y = -56 = 2x
![Page 25: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/25.jpg)
2 = 2
z = 4x – y
= 4.3 – (-8)
z = 12 + 8 = 20
x + y = -5
3 + y = -5
y = -5 - 3 = -8
6 = 2x
x = 6/2 = 3
![Page 26: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/26.jpg)
205
652
B206
2A
2.36
20 4.3 – (-8)12 + 820
3 + (-8)-5
y4x2x52
Bz6yx2
A
![Page 27: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/27.jpg)
1. PENJUMLAHAN DAN PENGURANGAN MATRIKS
Dua atau lebih matriks dapat dijumlahkan atau dikurang kan jika :
a. Matriks tersebut berordo sama
b. Yang dioperasikan elemen yang seletak
![Page 28: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/28.jpg)
Contoh:
215
49
942
7010
8122
536
CBA
Dapatkah A dan C dijumlahkan?
Jika
Dapatkah A dan B dijumlahkan?
![Page 29: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/29.jpg)
1164
12316
942
7010
8122
536
A + B = …
Untuk
942
7010
8122
536
BA
1780
234
8122
536
942
7010
B - A = …
![Page 30: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/30.jpg)
2. PERKALIAN MATRIKS
a. Perkalian 2 buah matriks
=
![Page 31: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/31.jpg)
tsrqpo
nm
lkj
ihg
fed
cba
CBA321
3 x 3 3 x 2 2 x 4
1. Dapatkah A dan C dikalikan?
2. Dapatkah A dan B dikalikan?
![Page 32: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/32.jpg)
Contoh
05
20
12/1
43
802
536
CA
Dapatkah A dan C dikalikan?
Diberikan
A 3 x 2 C2 x 4
=
C2 x 4
Z3 x 4
![Page 33: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/33.jpg)
05
20
12/1
43
802
536
CA
A x C = …
Untuk
A 3 x 2 C2 x 4
=
![Page 34: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/34.jpg)
K1C K2C K3C K4C
B1A
B2A
B3A
![Page 35: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/35.jpg)
K1C K2C K3C K4C
B1A
B2A
B3A
05
20
12/1
43
802
536
CAa = (6x3)+(2x4)
= 18 + 8
= 26
![Page 36: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/36.jpg)
K1C K2C K3C K4C
B1A
B2A
B3A
05
20
12/1
43
802
536
CAa = (-3x3)+(0x4)
= -9 + 0
= -9
26
![Page 37: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/37.jpg)
K1C K2C K3C K4C
B1A
B2A
B3A
05
20
12/1
43
802
536
CAa = (5x5)+(0x-8)
= 25 + 0
= 25
26
-9
![Page 38: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/38.jpg)
K1C K2C K3C K4C
B1A
B2A
B3A
05
20
12/1
43
802
536
CA
26
-925
![Page 39: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/39.jpg)
26
-925
05
20
12/1
43
802
536
CA
-17 10,5
-1,51 -4
016
30
-15A.C =
![Page 40: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/40.jpg)
Kerjakan soal berikut!
232
140
421
42
53
21
ZX Y
Diberikan
Tentukanlah matriks :
1.X.Y
2.Z.X
1.
![Page 41: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/41.jpg)
b. Perkalian Matriks dengan skala
Multiplication a real number with matrix A is multipilcation each elements of matrix A by that real number
k.A = [k.amn]
![Page 42: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/42.jpg)
Example
802
536
A
Determine 2 x A if
![Page 43: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/43.jpg)
Answer
2 x 82 x 02 x 2
5 x 23 x2
6 x 2 2.A =
1604
10612
=
![Page 44: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/44.jpg)
DETERMINANT
Determinant of matrix
a.Only used in square
b.are substraction with elements 1st diagonal and 2nd diagonal, where each elements enclosed
![Page 45: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/45.jpg)
a. DETERMINANT ORDO 2 X 2
If
dcba
A
than|A| = ad - bc
![Page 46: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/46.jpg)
Example
Determine value of determinant matrix below
61-105
A
Answer:
|A| = 5.6 – 10.-1 = 30 + 10 = 40
![Page 47: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/47.jpg)
DETERMINAN ORDO 3 x 3
If given
ihgfedcba
A
than |A| =
heb
gda
ihgfedcba
A
![Page 48: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/48.jpg)
DETERMINAN ORDO 3 x 3
|A| =
heb
gda
ihgfedcba
A
= (a.e.i + b.f.g + c.d.h) –(c.e.g + a.f.h + b.d.i)
![Page 49: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/49.jpg)
Example
Determine determinat of
531312740
A
Answer: = (0.1.5 + 4.-3.-1 + 7.2.3) –(-1.1.7 +3.-3.0 + 5.2.4)
= (0+12+42) – (-7+0+40)
= 54 – 33 = 21
314
120
531312740
A
![Page 50: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/50.jpg)
4. ADJOINAdjoin matrix A is the result transpose from kofaktor matriks A.
Matrix A
Adjoin Matrix A
Minor Matrix A
Kofaktor Matrix A
![Page 51: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/51.jpg)
Minor Jika maka minor
61-105
A
M11 = 6
61-105
AM12 = -1
61-105
A
M21 = 10
61-105
AM22 = 5
61-105
A
5101-6
A
a. Ordo 2 x 2
![Page 52: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/52.jpg)
Kofactor
If than kofactor
61-105
A
M11 = 6 .-11+1 = 6M12 = -1. -11+2 = -1.-1 =1M21 = 10. -12+1 = 10. -1 = -10M22 = 5. -1 2+2 = 5.1 = 5
510-16
A
5101-6
A
-
-
![Page 53: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/53.jpg)
Adjoin
If than Adjoin matrix A
Resulted from the its kofactor
61-105
A
510-16
Akofaktor
5101-6
AMinor
5110-6
A Adjoin
![Page 54: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/54.jpg)
205321211
A
M11 = 2.-2 – (0.3)
= -4- 0
= -4
205321211
A
M12 = 1.-2 – (-5.3)
= -2 – (-15)
= 13
205321211
A
M13 = 1.0 – (-5.2)
= 0 – (-10)
= 10
If , minor matrix A showed
next
205321211
A
b. Ordo 3 x 3
![Page 55: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/55.jpg)
205321211
A
M21 = 1.-2 – 0.2
= -2- 0
= -2
205321211
A
M22 = 1.-2 – (-5.2)
= -2 – (-10)
= 8
205321211
A
M23 = 1.0 – (-5.1)
= 0 – (-5)
= 5
![Page 56: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/56.jpg)
205321211
A
M21 = 1.3 – (2.2)
= 3 - 4
= -1
205321211
A
M22 = 1.3 – (1.2)
= 3 – 2
= 1
205321211
A
M23 = 1.2 – (1.1)
= 2 – (1)
= 1
![Page 57: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/57.jpg)
Kofactor
11158210134-
A
:Minor
A
11-15-82
1013-4-A
:Kofactor
![Page 58: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/58.jpg)
Adjoin
111582111-
:AMinor
11-15-8211-1-
:AKofactor
15-11-81-1-21-
:A Adjoin
205321211
A
given If
![Page 59: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/59.jpg)
Inverse matrix A AAdjoin.|A|
1A 1 0 A ,
5. INVERSE
![Page 60: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/60.jpg)
a. Inverse ordo 2 X 2
,
dcba
AIf
acbd
|A|1Aor 1
AAdjoin.|A|
1A 1
![Page 61: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/61.jpg)
Answer
dcba
Aif
acbd
|A|1Ao 1r
AAdjoin.|A|
1A , 1
Contoh:
61-105
A
Determine inverse from
![Page 62: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/62.jpg)
Answer :
51106
|)10.1(6.5|11A
51106
|1030|11A
51106
4011A
.adjA|A|
1A 1
![Page 63: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/63.jpg)
40/540/140/1040/61A
8/140/14/120/31A
![Page 64: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/64.jpg)
II. MATRIX APPLICATIONUsing to determine variabel value of linear equation. If the equation have variabel x dan y, than ..
|A||X|x
|A||Y|y
![Page 65: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/65.jpg)
Example Determine value of x dan y from the next
equations2x + 3y = 7x - 2y = 7
7
72132
yx
734(1.3)22.|A|21
32A
35211
4(7.3)27.|X|27
37X
774(1.7)2.|Y|1
72Y
17
7
![Page 66: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/66.jpg)
5735
|A||X|x
17 -7|A|
|Y|y
![Page 67: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/67.jpg)
Competence Check1. Given
(A.B)-1 = ….
455-6-
Ba4321
A nd
42-3-1
b.
2-1211
21
e.
21211-
21
c.
21-211-
21
d.
1234
a.
![Page 68: Materi Matriks SMK AP](https://reader034.vdocuments.pub/reader034/viewer/2022042421/577cc77c1a28aba711a113e3/html5/thumbnails/68.jpg)
2. Determine solution set from the next l
are ….
54
yx
2132
)2,1( d.)2,1(b.
)1,2(e.)2,1( c.)2,1(a.
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204321301
A
b. Find determinan and adjoint from the next matrix
3-1-12
B
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3-1-12
B
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Minor Jika maka minor
3-1-12
A
M11 = -3
3-1-12
AM12 = -1
3-1-12
A
M21 = 1
3-1-12
AM22 = 2
3-1-12
A
211-3-
A
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Kofactor
If than kofactor
3-1-12
A
M11 = -3 .-11+1 = -3M12 = -1. -11+2 = -1.-1 =1M21 = 1. -12+1 = 1. -1 = -1M22 = 2. -1 2+2 = 2.1 = 2
2-1-13-
A
211-3-
A
-
-
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SOAL
3-1-12
A
2-1-13-
A
211-3-
A
2
31
1-
MINOR
KOFACTOR
ADJOINT