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MATRIKS
MATRIKS
*
M
bxl
Bentuk Matriks
Matriks Tak Segi
Matriks Segi
(m = n)
B=
b11
b21
b31
.
..
.
bm1
b12
b22
b32
.
.
.
bm2
b13 …………
b23 …………
b33 …………
.
.
.
bm3 ………..
b1n
b2n
b3n
.
.
.
bmn
Matriks Segi
M3 =
1
6
9
0
4
8
5
2
8
Matriks setangkup
S4 =
A4 =
2 -1 3 1
3 4 0 0
9 5 2 7
8 1 4 -6
Matriks miring setangkup
2
3
3 2
9 0
5
1
9
5
0 6 4
1 4 7
M4 =
0 -1 2 -2
1 0 6 8
-2 -6 0
2 -8 -5
5
0
Matriks diagonal
D4 =
2 0 0
0 -8 0
0
0
Matriks tanda
0
0
1
0
4
3
2 1
7 2
0
0
0
0
2 4
0 9
1 0
0 -1
0
0 1 0
0 0 5
Matriks segitiga atas
A4 =
T3 =
0
0
0 -1
Matriks segitiga bawah
B4 =
1
4
0
3
0 0
0 0
2
1
7
2
2 0
4 9
Matriks nol
N3 =
0
0
0
0
0
0
0
0
0
Matriks satu
S3 = 1 1 1
Matriks satu-nol
M3 =
Matriks skalar
S4 =
8
0
0
0
0
8
0
0
0
0
8
0
1
0
0
1
1
1
0
1
0
1
1
1
1
1
1
Matriks Identitas
0
0
0
8
I4 =
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
Matriks Tak Segi
Matriks datar
W =
3x4
1 6 5 2
9 0 2 4
4 8 8 7
Matriks tegak
M = 1 9 4
4x3
6 0 8
5 2 8
2 4 7
Matriks nol
X =
4x3
0
0
0
0
0
0
0
0
0
0
0
0
E =
2x3
0 0 0
0 0 0
Matriks satu
M =
4x2
1
1
1
1
S =
1
1
1
1
3x4
1
1
1
1
1
1
1
1
1
1
1
1
Matriks satu-nol
F =
4x3
1 0
0 1
1 0
1 1
1
1
0
1
Y =
3x4
0
1
1
0 1
0 1
1 0
1
0
1
Penjumlahan 2 Matriks
Hanya berlaku bila :
A
B
ba x la
( ba = bb
A = a11 a12 a13
4x3
a21 a22 a23
a31 a32 a33
a41 a42 a43
bb x lb
&
la = lb )
B = b11 b12 b13
4x3
b21 b22 b23
b31 b32 b33
b41 b42 b43
* Tambah
a11+b11 a12+b12 a13+b13
A+B =
(4 x 3)
A =
2
3
a21+b21 a22+b22 a23+b23
a31+b31 a32+b32 a33+b33
a41+b41 a42+b42 a43+b43
B =
5
2x3
(2 x 3)
6
1
2
0
1
5
9
6
5
0
0
2x3
3
A+B =
3
0
-1
2+3
3+6
5+1
3+2
0+0
-1+1
=
* Kurang
A-B =
(4 x 3)
A =
2
3
a11-b11 a12-b12 a13-b13
a21-b21 a22-b22 a23-b23
a31-b31 a32-b32 a33-b33
a41-b41 a42-b42 a43-b43
B =
5
2x3
(2 x 3)
6
1
2
0
1
-1
-3
4
1
0
-2
2x3
3
A-B =
3
0
-1
2-3
3-6
5-1
3-2
0-0
-1-1
=
Penggandaan 2 Matriks
Hanya berlaku bila :
A
ba x la
( bb
A = a11 a12
4x2
a21 a22
a31 a32
a41 a42
x
B
bb x lb
= la)
=
C
ba x lb
B = b11 b12 b13
2x3
b21 b22 b23
AxB =
(4 x 3)
C
a11b11+a12b21
a21b11+a22b21
a31b11+a32b21
a41b11+a42b21
(4 x 3)
=
c11+c11
c21+c21
c31+c31
c41+c41
a11b12+a12b22 a11b13+a12b23
a21b12+a22b22 a21b13+a22b23
a31b12+a32b22 a31b13+a32b23
a41b12+a42b22 a41b13+a42b23
c12+c12
c22+c22
c32+c32
c42+c42
c13+c13
c23+c23
c33+c33
c43+c43
A =
2
3
2x3
A x B =
2x3
3x2
=
=
3
0
2
3
B =
5
-1
3
0
3
2
6
0
1
1
3x2
5
-1
3
2
6
0
1
1
(2)(3)+(3)(6)+(5)(1)
(2)(2)+(3)(0)+(5)(1)
(3)(3)+(0)(6)+(-1)(1)
(3)(2)+(0)(0)+(-1)(1)
29
9
8
5
B x A =
3x2
3
2
2
3
5
6
0
3
0
-1
1
1
2x3
=
(3)(2)+(2)(3) (3)(3)+(2)(0) (3)(5)+(2)(-1)
(6)(2)+(0)(3) (6)(3)+(0)(0) (6)(5)+(0)(-1)
(1)(2)+(1)(3) (1)(3)+(1)(0) (1)(5)+(1)(-1)
=
12 9
12 18
5
3
13
30
4
Putaran Suatu Matriks
3x2
3x4
2 4 1 4
3 0 0 7
4 9 5 1
M’ = m11 m21 m31
2x3

M =
m21 m22
m31 m32

M = m11 m12
M’ = (m’ji)lb

M = (mij)bl
m12 m22 m32
M =
4x3
2
4
1
4
3
0
0
7
4
9
5
1
Teras Suatu Matriks
M = (mij)bb
tr M =
M3 =
M3 =
mii = m11 + m22 + m33 ……….. + mbb
m11 m12 m13
m21 m22 m23
m31 m32 m33
2
3
6
5
6
1
0
9
4
tr M = m11 + m22 + m33
tr M = 2 + 6 + 4
= 12
Matriks Sekatan
• Pengolahan ganda pada 2 buah matriks yang
berdimensi (ukuran) besar biasanya sulit
dilakukan. Untuk memudahkannya dilakukan
penyekatan sehingga terbentuk anak-anak
matriks dengan dimensi yang lebih kecil.
• Cara penyekat harus memperhatikan ketentuan
bahwa banyak jalur pada anak-matriks yang
digandakan harus samadengan banyaknya baris
anak-matriks pengganda.
M =
bxl
=
m11 m12 m13
m21 m22 m23
m14 ………… b1l
m24 ………… b2l
m31
.
..
.
mb1
m34 ………… b3l
.
.
.
.
.
.
mb4 ……….. mbl
m32 m33
.
.
.
.
.
.
mb2 mb3
M11
(p x q)
M21
(b – p)q
M12
p(l – q)
M22
(b – p)(l – q)
N =
lxk
=
n11 n12
n21 n22
n31 n32
n13
n23
n33
………… n1k
………… n2k
………… n3k
n41 n42
.
.
.
.
.
.
nl1 nl2
n43
.
.
.
nl3
………… n4k
.
.
.
………... nlk
N11
(q x r)
N21
(l – q)r
N12
q(k – r)
N22
(l – q)(k – r)
M
bxl
C =
bxk
x
N
lxk
=
C
bxk
M11 N11 + M12 N21
M11 N12 + M12 N22
M21 N11 + M22 N21
M21 N12 + M22 N22
(p x r)
(b – p)r
p(k-r)
(b – p)(k – r)
Transformasi Dasar
(pengolahan baris atau lajur terhadap suatu matriks
dengan cara
pertukaran letak, penjumlahan atau penggandaan)
A =
a11
a21
a31
.
.
.
am1
a12
a22
a32
.
.
.
am2
a13
a23
a33
.
.
.
am3
………… a1n
………… a2n
………… a3n
.
.
.
……….. amn
x =
2
2
1
2
1
3
4
3
2
4
6
2
• Pertukaran letak
A
E1.2
1
3
4
2
1
2
2
4
6
A
F1.2
1
2
2
3
1
4
4
2
6
• Penjumlahan
A
E3.2(1)
2
1
2
1
3
4
Brs 3 : 2
Brs 2 x 1 : 1
3
7 10
3
4
3
6
4
7
10
• Tambah
Ljr 3
A
F3.2(1)
Ljr 2 x 1
2
1
3
2
1
3
1
3
7
4
3
7
2
4 10
6
4
10
+
+
A
E3.2(-1)
2
1
1
3
2
4
1
1
2
Brs 3 :
2
4
6
Brs 2 x (-1) : -1
-3
-4
1
1
2
• Kurang
Ljr 3
2
A
F3.2(-1) 1
1
1
3
1
2
4
2
Ljr 2 x (-1)
2
-1
1
4
-3
1
6 -4
2
+
+
• Penggandaan
• Kali
A
E3(2)
2
1
4
1
3
8
8 12
2
4 12
2
1
2
2
1
1
1
3
4
1
3
2
1
2
3
2
4
3
2
1
2
1
3
4
4
A
F3(2)
• Bagi
A
E3(1/2)
A
F3(1/2)

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