matriks putta
TRANSCRIPT
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MATRIKS
PROGRAM PASCA SARJANA PENDIDIKAN MATEMATIKA UNSRI
YUDI YUNIKA PUTRA
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MATRIKS PENGERTIAN MATRIKS
BENTUK UMUM
OPERASI ALJABAR MACAM-MACAM MATRIKS
DETERMINAN TRACE
INVERS MATRIKS SOAL – SOAL MATRIKS
NOTASI MATRIKS ORDO MATRIKS
JENIS MATRIKS
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PENGERTIAN MATRIKS
MATRIKS ADALAH KUMPULAN BILANGAN YANG DINYATAKAN DALAM BARIS DAN KOLOM
BACK
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JENIS MATRIKS
.
MATRIKS PERSEGI
MATRIKS DIAGONAL
MATRIKS SATUAN
MATRIKS NOL
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OPERASI ALJABAR
.
PENJUMLAHAN DAN PENGURANGAN
PERKALIAN
BACK
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MACAM-MACAM MATRIKS
.
MATRIKS IDENTITAS (I)
TRANSPOSE ( )
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DETERMINAN
.
MATRIKS ORDO 2
MATRIKS ORDO 3
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TRACE
.
DEFINISI
SIFAT TRACE
BACK
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INVERS
.
PENGERTIAN
SINGULAR DAN NON SINGULAR
SIFAT INVERS
BACK
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SOAL – SOAL MATRIKS
.SOAL BAHAS UJIAN
NASIONAL
SOAL BAHAS UJIAN MASUK PTN
SOAL PENDALAMAN
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MATRIKS PERSEGI
Matriks persegi adalah
suatu matriks dimana
banyaknya baris sama dengan banyaknya kolom
987
654
321
A
BACK
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MATRIKS DIAGONAL
Matriks diagonal adalah
suatu matriks persegi
dengan setiap elemen yang tidak terletak pada diagonal utama adalah nol, sedangkan elemen-elemen pada diagonal utama tidak semuanya nol.
500
030
001
A
BACK
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MATRIKS SATUAN
Matriks satuan adalah
matriks diagonal dengan
setiap elemen diagonal
utama adalah 1
100
010
001
2I
10
011I
BACK
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MATRIKS NOL
Matriks Nol (0), yaitu
matriks yang semua elemennya bernilai 0
00
001A
000
000
000
2A
BACK
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CONTOH
PENJUMLAHAN & PENGURANGAN MATRIKS
.
Dua buah matriks bisa dijumlahkan atau dikurangkan, jika
1. Mempunyai Ordo sama
2. Dilakukan operasi elemen seletak
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hg
fe
dc
ba ea fb
gc hd
BACK
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PERKALIAN MATRIKS
SKALAR X MATRIKS
MATRIKS X MATRIKS
PERPANGKATAN
SIFAT
BACK
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PERKALIAN SKALAR DENGAN MATRIKS
dc
baK.
BACK
ka kbkc kd
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Matriks x Matriks
dhcfdgce
bhafbgaekj
hg
fek
dc
baj
.
BACK
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Perpangkatan
1
34
23
2
.
.
.
.
nn AAA
AAA
AAA
AAA
BACK
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NOTASI MATRIKS
Kurung biasa
Kurung sikuKurung doub
mutlak
BACK
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Sifat
AA
AA
ApqqAp
pBpABAp
qApAAqp
1
1
)()(
)(
)(
BACK
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BENTUK UMUM
n
n
aaa
aaaA
22221
11211
...
... Baris
Kolom
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BENTUK UMUM
n
n
aaa
aaaA
22221
11211
...
... Baris
KolomKeterangan :
a11: Elemen baris pertama kolom pertama
BACK
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ORDO MATRIKS
ORDO = banyak baris x banyak kolom
Contoh :
61
23A
Baris 1
Baris 2
Kolo
m 1
Kolo
m
2
Matriks A mempunyai ordo = 2x2
Ditulis : A2x2
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MACAM-MACAM MATRIKS
Matriks Identitas MATRIKS TRANSPOST TA
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Matriks identitas (i) merupakan matriks bujur sangkar yang elemen diagonal utama merupakan angka 1 dan selain itu angka nol
100
010
001
,10
0121 II
EXAMPLE
BACK
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Matriks transpost ( ) merupakan matriks yang diperoeh dengan menguba baris (matriks asal menjadi kolom atau kolom(matriks asal)menjadi baris
TA
ifc
heb
gda
A
ihg
fed
cba
A
cb
daA
cd
baA
t
t
,
,
BACK
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DETERMINAN
ORDO 2 ORDO 3
BACK
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ORDO 2
bcaddc
baA
cd
baA
BACK
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ORDO 3
)(( idbhfageccdhbfgaei
hg
ed
ba
ihg
fed
cba
A
ihg
fed
cba
A
BACK
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TRACE
Sama halnya dengan determinan, trace hanya didefenisikan pada matriks persegi, dinotasikan dengan Tr(A), yaitu jumlah elemen utama matrik A
ieaATr
ihg
fed
cba
A
)(
BACK
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SIFAT TRACE
)()( BATrABTr
)()( ATrATr T
)(.).( ATrpApTr
)()()( BTrATrBATr
BACK
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INVERS
ac
bd
bcadA
dirumuskan
AndinotasikaAinversmakadc
baA
1
:
,
1
1
)(11 AAdjA
A Ordo 3 x 3
Ordo 2 x 2
BACK
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CONTOH SOAL PENDALAMAN
1. Diketahui
A. p =1 dan q = -2
B. p =1 dan q = 2
C. p =-1 dan q = 2
D. p =1 dan q = 8
E. p = 5 dan q = 2
....,37
24
55
24maka
qqp
BACK
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Singular Matriks dinamakan singular bila det A = 0
BACK
Non Singular Matriks dinamakan singular bila det 0A
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Sifat – Sifat Invers
BACK
AA 11)(111)( ABAB
1111)( ABCABC
111 AAAA
AAA 11
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Ujian Nasional 2007
Diketahui Matriks
Apabila Maka nilai
a. 10b. 15c. 20d.25e. 30
BACK
13
27,
3
2,
41
12C
y
yxBA
TCAB ...xy
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Matematika Dasar SNMPTN 2010
Diketahui M adalah Matriks sehingga
maka determinan matriks M adalah . . . a. -2b. -1c. 0d. 1e. 2
BACK
dc
dbca
dc
baM .