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# MATRIKS - PowerPoint PPT Presentation

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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*

M

b x l

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 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 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 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

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

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

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)

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

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

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

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

• 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)

=

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

(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 =

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

• 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

+

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

+

• 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