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MATRIKS. MATRIKS. *. M b x l. Bentuk Matriks. Matriks Tak Segi. Matriks Segi. (m = n). b 11 b 12 b 13 ………… b 1n b 21 b 22 b 23 ………… b 2n b 31 b 32 b 33 ………… b 3n . . . . . . . . .

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Matriks1
MATRIKS

*

M

b x l


Bentuk matriks
Bentuk Matriks

Matriks Tak Segi

Matriks Segi

(m = n)

b11 b12 b13 ………… b1n

b21 b22 b23 ………… b2n

b31 b32 b33 ………… b3n

. . . .

. . . . .

. . . .

bm1 bm2 bm3 ……….. bmn

B =


A4 =

M3 =

  • 9 4

  • 6 0 8

  • 5 2 8

2 -1 3 1

3 4 0 0

9 5 2 7

8 1 4 -6

Matriks Segi

Matriks setangkup

Matriks miring setangkup

S4 =

M4 =

2 3 2 5

3 9 0 1

9 0 6 4

5 1 4 7

0 -1 2 -2

1 0 6 8

-2 -6 0 5

2 -8 -5 0


Matriks diagonal

Matriks tanda

D4 =

T3 =

1 0 0

0 -1 0

0 0 -1

2 0 0 0

0 -8 0 0

0 0 1 0

0 0 0 5

Matriks segitiga atas

Matriks segitiga bawah

A4 =

B4 =

1 4 2 1

0 3 7 2

0 0 2 4

0 0 0 9

1 0 0 0

4 3 0 0

2 7 2 0

1 2 4 9


Matriks nol

Matriks satu

S3 =

N3 =

1 1 1

1 1 1

1 1 1

0 0 0

0 0 0

0 0 0

Matriks satu-nol

M3 =

1 0 1

0 1 1

0 1 0

Matriks skalar

Matriks Identitas

I4 =

S4 =

8 0 0 0

0 8 0 0

0 0 8 0

0 0 0 8

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1


Matriks datar

Matriks tegak

Matriks Tak Segi

W =

3 x 4

M =

4 x 3

  • 9 4

  • 0 8

  • 5 2 8

  • 2 4 7

1 6 5 2

9 0 2 4

4 8 8 7

Matriks nol

X =

4 x 3

E =

2 x 3

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0


Matriks satu

M =

4 x 2

  • 1

  • 1 1

  • 1 1

  • 1 1

S =

3 x 4

1 1 1 1

1 1 1 1

1 1 1 1

Matriks satu-nol

F =

4 x 3

Y =

3 x 4

  • 0 1

  • 0 1 1

  • 1 0 0

  • 1 1 1

0 0 1 1

1 0 1 0

1 1 0 1


Penjumlahan 2 matriks
Penjumlahan 2 Matriks

Hanya berlaku bila :

A

ba x la

B

bb x lb

( ba = bb & la = lb )

A =

4 x 3

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43

B =

4 x 3

b11 b12 b13

b21 b22 b23

b31 b32 b33

b41 b42 b43


A+ B =

(4 x 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

* Tambah

3 6 1

2 0 1

2 3 5

3 0 -1

A =

2 x 3

B =

2 x 3

A+ B = =

(2 x 3)

2+3 3+6 5+1

3+2 0+0 -1+1

5 9 6

5 0 0


A - B =

(4 x 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

* Kurang

3 6 1

2 0 1

2 3 5

3 0 -1

A =

2 x 3

B =

2 x 3

A- B = =

(2 x 3)

2-3 3-6 5-1

3-2 0-0 -1-1

-1 -3 4

1 0 -2


Penggandaan 2 matriks
Penggandaan 2 Matriks

Hanya berlaku bila :

A x B = C

ba x labb x lb ba x lb

( bb = la)

A =

4 x 2

a11 a12

a21 a22

a31 a32

a41 a42

B =

2 x 3

b11 b12 b13

b21 b22 b23


Ax B =

(4 x 3)

a11b11+a12b21 a11b12+a12b22 a11b13+a12b23

a21b11+a22b21 a21b12+a22b22 a21b13+a22b23

a31b11+a32b21 a31b12+a32b22 a31b13+a32b23

a41b11+a42b21 a41b12+a42b22 a41b13+a42b23

c11+c11 c12+c12 c13+c13

c21+c21 c22+c22 c23+c23

c31+c31 c32+c32 c33+c33

c41+c41 c42+c42 c43+c43

C =

(4 x 3)


3 2

6 0

1 1

B =

3 x 2

A =

2 x 3

2 3 5

3 0 -1

3 2

6 0

1 1

A x B =

2 x 3 3 x 2

2 3 5

3 0 -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)

=

  • 9

  • 8 5


3 2

6 0

1 1

2 3 5

3 0 -1

B x A =

3 x 2 2 x 3

(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 13

12 18 30

5 3 4


Putaran suatu matriks
Putaran Suatu Matriks

M = (mij)bl M’ = (m’ji)lb

M =

3 x 2

m11 m12

m21 m22

m31 m32

m11 m21 m31 m12 m22 m32

M’ =

2 x 3

M =

4 x 3

  • 2 3 4

  • 09

  • 1 05

  • 47 1

  • 41 4

  • 00 7

  • 4 95 1

M =

3 x 4


Teras suatu matriks
Teras Suatu Matriks

M = (mij)bb

tr M = mii = m11 + m22 + m33 ……….. + mbb

M3 =

m11 m12 m13

m21 m22 m23

m31 m32 m33

tr M = m11 + m22 + m33

M3 =

2 5 0

3 69

6 14

tr M = 2 + 6 + 4

= 12


Matriks sekatan
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.


m11 m12 m13 m14 ………… b1l

m21 m22 m23 m24 ………… b2l

m31 m32 m33 m34 ………… b3l

. . . . .

. . . . . .

. . . . .

mb1 mb2 mb3 mb4 ……….. mbl

M =

b x l

M11 M12

(p x q) p(l – q)

M21 M22

(b – p)q (b – p)(l – q)

=


n11 n12 n13 ………… n1k

n21 n22 n23 ………… n2k

n31 n32 n33 ………… n3k

n41 n42 n43 ………… n4k

. . . .

. . . .

. . . .

nl1 nl2 nl3 ………... nlk

N =

l x k

N11 N12

(q x r) q(k – r)

N21 N22

(l – q)r (l – q)(k – r)

=


M x N = C

b x l l x k b x k

M11 N11 + M12 N21 M11 N12 + M12 N22

(p x r) p(k-r)

M21 N11 + M22 N21 M21 N12 + M22 N22

(b – p)r (b – p)(k – r)

C =

b x k


Transformasi dasar
Transformasi Dasar

(pengolahan baris atau lajur terhadap suatu matriks

dengan cara

pertukaran letak, penjumlahan atau penggandaan)

a11 a12 a13 ………… a1n

a21 a22 a23 ………… a2n

a31 a32 a33 ………… a3n

. . . .

. . . .

. . . .

am1 am2 am3 ……….. amn

A =


Pertukaran letak
Pertukaran letak

x =

2

3

2

  • 1 2

  • 1 3 4

  • 2 4 6

1 2 2

3 1 4

4 2 6

1 3 4

2 1 2

2 4 6

E1.2

F1.2

A

A


Penjumlahan
Penjumlahan

  • 1 2

  • 1 3 4

  • 3 7 10

Brs 3 : 2 4 6

  • Tambah

E3.2(1)

Brs 2 x 1 : 1 3 4

A

+

3 7 10

Ljr 3

Ljr 2 x 1

  • 1 3

  • 1 3 7

  • 2 4 10

2

4

6

3

7

10

1

3

4

F3.2(1)

A

+


2 1 2

1 3 4

1 1 2

Brs 3 : 2 4 6

E3.2(-1)

A

Brs 2 x (-1) : -1 -3 -4

+

  • Kurang

1 1 2

Ljr 3

Ljr 2 x (-1)

  • 1 1

  • 1 3 1

  • 2 4 2

2

4

6

1

1

2

-1

-3

-4

F3.2(-1)

A

+


Penggandaan
Penggandaan

  • K a l i

  • 1 4

  • 1 3 8

  • 2 4 12

  • 1 2

  • 1 3 4

  • 4 8 12

E3(2)

F3(2)

A

A

  • B a g i

  • 1 1

  • 1 3 2

  • 2 4 3

  • 1 2

  • 1 3 4

  • 1 2 3

E3(1/2)

F3(1/2)

A

A


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