BAB XIX. MATRIKS - 19. Matriks

  B. Kesamaan dua buah Matriks :

BAB XIX. MATRIKS

  A = B

  Pengertian: a b p q a = p , b = q

  ⎛ ⎞ ⎛ ⎞ Matriks adalah susunan bilangan-bilangan yang diatur

  = ⇔ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  c d r s c = r , d = s

  ⎝ ⎠ ⎝ ⎠ pada baris dan kolom dan letaknya di antara dua buah kurung.

A. Operasi Matriks :

  C. Determinan Matriks :

  1. Matriks ordo 2 x 2 2 baris dan 2 kolom Baris

  a b p q

  ⎛ ⎞ ⎛ ⎞ Jika A = dan B =

  ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ a b ⎛ ⎞

  c d r s

  ⎝ ⎠ ⎝ ⎠ Jika A =

  ⎜⎜ ⎟⎟

  c d

  ⎝ ⎠ kolom Maka det(A) = |A| = ad – bc Æ jika det(A) = 0 maka disebut matriks berordo 2x2 matriks A disebut matriks singular

  1. Penjumlahan

  2. Matriks ordo 3 x 3

  a b p q a p b q

• ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

  A + B = = + ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

• c d r s c r d s ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

  a b c ⎛ ⎞

  2. Pengurangan

  ⎜ ⎟

  Jika A = d e f

  ⎜ ⎟ ⎜ ⎟ a b p q a − p b − q g h i

  ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

  ⎝ ⎠

  A – B = = - ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  c d r s c − r d − s

  ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Maka det(A) = |A| = aei + bfg + cdh – gec – hfa – idb diagram :

  3. Perkalian

• arah negatif

  a b c a b a.

  Perkalian skalar

  d e f d e a b ka kb

  ⎛ ⎞ ⎛ ⎞

  g h i g h

  k = ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  c d kc kd

  ⎝ ⎠ ⎝ ⎠ + + + arah positif

  b. Perkalian matriks dengan matriks

  D. Invers Matriks : a b p q

  ⎛ ⎞ ⎛ ⎞

  1 ⎛ ⎞

  A . B =

• Jika A.B = I ; I = , maka A dan B dikatakan ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  ⎜⎜ ⎟⎟

  c d r s

  1 ⎝ ⎠ ⎝ ⎠

  ⎝ ⎠ saling invers

• ap br aq bs ⎛ ⎞

  = ⎜⎜ ⎟⎟

  b

  ⎛ ⎞ ⎛ ⎞

  − ¹ cp dr cq ds

• a b d −
• 1

  ⎝ ⎠

• Jika A = , maka A = . =

  ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  c d A − c a

  det( ) ⎝ ⎠ ⎝ ⎠

  d − b

  ⎛ ⎞

  1 .

  ⎜⎜ ⎟⎟

  ad − bc − c a

  ⎝ ⎠

  www.belajar-matematika.com - 1

19. SOAL-SOAL MATRIKS

  14

  = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − + − − −

  p p

  4 32 .

  3

  4

  24

  2 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  8

  24

  4

  2

  3. Diketahui matriks A= ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2

  3

  5

  8 , B =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  2

  3

  2

  p p

  4 3 ). ) 2 ( 1 .(

  dan C =

  2

  8

  1 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  8

  14

  24

  2 ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  p

  4

  4 ’

  2 8 .

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  4

  3

  8

  1 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − + + − − − + − − + − −

  ) 4 .( 8 .

  4 ) 3 . 1 .(

  4 ) 4 .(

  x

  ⎟⎟ ⎠ ⎞

  4

  3 3 ). . 2 (

  − −

  2

  3

  5

  8 .

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  2

  3 2 x =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − + − + − + − + 2 ).

  2 ( 2 .

  3 2 ).

  ⎟⎟ ⎠ ⎞

  5 ( 2 .

  8 3 ). . 5 (

  8

  x x

  = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2 6 .

  3

  6 15 .

  8

  x x

  ….(1)

  ⎜⎜ ⎝ ⎛

  y

  ⎜⎜ ⎝ ⎛

  2

  3

  5

  3 9 y Jika matriks A.B = A + C, maka nilai x + y = …

  A. 2 B. 4 C. 5 D. 6 E. 8 jawab: A.B = A + C

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2

  3

  5

  8 .

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  3

  9

  2

  x

  = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2

  3

  5

  8

  ⎜⎜ ⎝ ⎛

  3

  5

  3

  3

  − −

• 4 + 3p = 14 32 – 4p = 8 3p = 18 32 – 8 = 4p = 6 24 = 4p p = 6 jawabannya adalah E UAN2004
• 3
• 3
• ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  7

  2

  1

  4 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  14

  8

  b a

  ⎟⎟ ⎠ ⎞

  − −

  ⎟⎟ ⎠ ⎞

  7

  2

  1

  4

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  7

  2

  1

  4 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  ⎜⎜ ⎝ ⎛

  4

  7

  1

  EBTANAS1998

  1. Diketahui matriks A = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  7

  2

  1

  4 ;

  B = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  7

  2

  4 dan C= ⎟⎟ ⎠ ⎞

  1

  ⎜⎜ ⎝ ⎛

  − −

  14

  8

  b a

  Nilai a dan b yang memenuhi A + 3B = C Berturut-turut adalah…

  A. 2 dan 4 C. -8 dan -14 E. 8 dan 14

  B. -2 dan 4 D. 8 dan -14 Jawab: A + 3B = C

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  7

  2

  − −

  2

  ⎜⎜ ⎝ ⎛

  2

  12

  1

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  p

  4

  4 .

  1

  ⎟⎟ ⎠ ⎞

  12 =

  3

  6

  21

  − −

  ⎜⎜ ⎝ ⎛

  4

  4 =

• − −
• 4

  ⎜⎜ ⎝ ⎛

  10 E. 6 Jawab : A.B = C

  24

  2 Jika AB=C, nilai p=…

  A. -6 B. -

  3

  10 C.

  3

  1 D.

  3

  3

  8

  2

  6

  7

  21

  − + −

  ⎜⎜ ⎝ ⎛

  ⎟⎟ ⎠ ⎞

• 4
• ⎟⎟ ⎠ ⎞

  14

  − −

  − −

  4

  14

  4

  2

  8 Didapat a = 2 dan b = 4 Jawabannya adalah A EBTANAS2000

  2. Diketahui matrik A = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  p

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  4 B= ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  4

  3

  8

  1 , dan C=

  ⎟⎟ ⎠ ⎞

  2

  2 …(2)

  12

  3

  6

• ⎟⎟ ⎠ ⎞

• − +
• − +

  1 6x+ 24 y = 0 6x = -24y 6x = -24 . (-

  1 =

  2

  4

  6x = 12 x = 2 x. y = 2. -

  1 )

  2

  (1) = (2)

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  1

  ⎟⎟ ⎠ ⎞

  2

  ⎜⎜ ⎝ ⎛

  − − − −

  10

  10

  2

  4

  12

  3

  6

  2 2x + 6y = 1 x 3 ⇒ 6x + 18y = 3 3x+12y = 0 x 2 ⇒ 6x+ 24 y = 0 - 0 - 6y = 3 y = -

  y x y x y x y x

  − − − −

  y x y x y x y x

  ⎜⎜ ⎝ ⎛

  2

  1

  3

  2 + y.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  10

  4

  12

  6 =

  ⎟⎟ ⎠ ⎞

  − − x x

  ⎜⎜ ⎝ ⎛

  x x

  2

  3

  2

  ⎜⎜ ⎝ ⎛

  − − y y

  y y

  10

  4

  12

  6 =

  ⎟⎟ ⎠ ⎞

  2

  UAN2004

  1 = - 1 jawabannya adalah B

  ⎜⎜ ⎝ ⎛

  36

  4

  8

  4 C.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  38

  4

  8

  4 jawab: Karena fungsi f(S,M) = S ² -M ² maka

  Fungsi f(S+M, S-M) = (S+M) ² - (S-M) ² S + M = ⎟⎟ ⎠ ⎞

  −

  ⎜⎜ ⎝ ⎛

  3

  1

  2

  ⎜⎜ ⎝ ⎛

  − 3

  2

  1 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  −

  1

  2

  − −

  ⎟⎟ ⎠ ⎞

  5. Diketahui matriks S = ⎟⎟ ⎠ ⎞

  4

  ⎜⎜ ⎝ ⎛

  3

  1

  2 dan M = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − 3

  2

  1 . Jika fungsi f(S,M) = S ² -M ² matriks f(S+M, S-M) adalah…

  A.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − 40

  20

  4 E.

  4 D.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − − −

  40

  4

  20

  4 B.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − 30

  4

  20

  − −

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  2

  3 17 y …(2)

  (1) = (2)

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2 6 .

  3

  6 15 .

  8

  x x

  = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  6

  2

  3 17 y 8x-15 = 17 3y = 6 8x = 32 y = 2 x = 4 x + y = 4 + 2 = 6 jawabannya adalah D EBTANAS2000

  4. Diketahui A = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2

  1

  3

  2 , B =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  10

  6

  ⎜⎜ ⎝ ⎛

  12

  9

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2

  3

  5

  8

  ⎜⎜ ⎝ ⎛

  4

  3

  5

  3

  y

  = ⎟⎟ ⎠ ⎞

  = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  4

  2

  3

  3

  5

  3

  5

  9

  8

  y

  4

  6 Dan A ² = x.A + y.B, nilai xy=…

  1 …(1) x.

  3

  6 ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2

  1

  3

  2 .

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2

  1

  2 =

  4

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − − + − − − + − − + − +

  ) 2 ).( 2 ( 3 . 1 ) 1 ).(

  2 ( 2 .

  1 ) 2 .(

  3 3 . 2 ) 1 .(

  3 2 .

  2 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  1

  12

  10

  A. -4 B. -1 C. -

  ⎜⎜ ⎝ ⎛

  2

  1 D. 1

  2

  1 E. 2 jawab: A ² = x.A + y.B

  ⇔ A. A = x.A + y.B ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2

  1

  3

  2 .

  ⎟⎟ ⎠ ⎞

  − −

  − −

  2

  1

  3

  2 = x.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  2

  1

  3

  2 + y.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  3

• ⎟⎟ ⎠ ⎞
• ⎟⎟ ⎠ ⎞

  ⎛

  3 2 ⎞ ⎛

  3 2 ⎞ ⎛

  7 6 ⎞

  1 −

  1 ₂ ⎛ ⎞

  2 − 2 −

  2

  2 ⎛ ⎞ ⎛ ⎞

  (S+M) = . = ⎜ ⎟

  1 A.

  C.

  E. ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  − − − − ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  1

  1

  3

  2

  −

  3

  4

  ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎜ ⎟ −

  7

  8 7 −

  8 ⎝ ⎠ ⎝ ⎠

  ⎝ 2 ⎠ 4 − 1 ⎛ ⎞

  − 2 −

  2 ⎛ ⎞

  2

  1

  2 2 − 1 −

  2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

  ⎜ ⎟

  1 B.

  D. S – M = =

  ⎜⎜ ⎟⎟

• ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ −

  3

  − 7 −

  8 −

  1 ⎜ ⎟

  1 3 − 3 − 1 − 3 − ( − 3 ) ⎝ ⎠

  ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

  ⎝ 2 ⎠

  1 −

  2 ⎛ ⎞

  = ⎜⎜ ⎟⎟

  Jawab: −

  1

  6 ⎝ ⎠

  M = A + B 1 − 2 1 −

  2 3 −

  14 ₂ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (S-M) = . =

  ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ −

  1 6 −

  1 6 −

  7

  38 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

  3 4 − 1 −

  2

  2

  2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

• = = ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  5

  1

  2

  7

  7

  8 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

  7

  6 3 −

  14

  4

  20 _{2 2} ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

• (S+M) - (S-M) = = ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  − 3 − 2 −

  7

  38 4 − 40 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ d b

  − 1 ⎛ ⎞

  − ¹ M = .

  ⎜⎜ ⎟⎟

  c a ad − bc −

  ⎝ ⎠ Jawabannya adalah A

  8 −

  2 ⎛ ⎞

  1 EBTANAS1997 =

  ⎜⎜ ⎟⎟ −

  7

  2 2 . 8 − 2 .

  7 − x

  ⎛ 3 ⎞ ⎝ ⎠ 6. Diketahui A = adalah matriks singular.

  ⎜⎜ ⎟⎟

  6

  8 ⎝ ⎠

  4 −

  8 −

  2 Nilai x = …. ⎛ ⎞

  1 ⎛ ⎞

  1

  ⎜ ⎟

  1

  = . = ⎜⎜ ⎟⎟

  −

  3

  2 −

  7

  2 ⎝ ⎠

  1 ⎜ ⎟

  A. -5 B. -4 C. -3 D. 3 E. 4 jawabannya adalah D Jawab: teori:

  UAN2007 − x y

  ⎛

  2 1 ⎞ ⎛ + 2 ⎞

  a b

  ⎛ ⎞

  8. Diketahui matriks A = ; B = Jika A =

  ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  y

  1

  4

  3 ⎝ ⎠ ⎝ ⎠

  c d

  ⎝ ⎠ ⎛

  7 2 ⎞ Maka det(A) = |A| = ad – bc Æ jika det(A) = 0 maka dan C =

  ⎜⎜ ⎟⎟ matriks A disebut matriks singular

  3

  1 ⎝ ⎠ _{t t} apabila B – A = C dan C = transpose matriks C,

  3 − x ⎛ ⎞ maka nilai x. y = …

  A = ⎜⎜ ⎟⎟

  6

  8 ⎝ ⎠

  A. 10. B. 15 C. 20 D. 25 E. 30 Det(A) = ad – bc = 3.8 – (-x).6 jawab: = 24 + 6x =0

  a b a c

  6x = -24 ⎛ ⎞ ⎛ ⎞ _t teori : Jika A = , maka A = ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ x = -4

  c d b d

  ⎝ ⎠ ⎝ ⎠ jawabannya adalah B

  7

  2

  7

  3 ⎛ ⎞ ⎛ ⎞ _t

  C = Æ C = ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  3

  1

  2

  1 UAN2006 ⎝ ⎠ ⎝ ⎠

  3 4 − 1 −

  2 ⎛ ⎞ ⎛ ⎞

  7. Diketahui matriks A= dan B = ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

  5

  1

  2

  7 ⎝ ⎠ ⎝ ⎠ ₁

  − jika M = A + B, maka invers M adalah M = ….

• ⎟⎟ ⎠ ⎞

  23

  3

  2

  1 dan AB ¹

  −

  = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  3

  4

  1

  2 , maka A =… A.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  13

  ⎜⎜ ⎝ ⎛

  9

  5 C.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  23

  9

  5

  3 E.

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  3

  12

  5

  5

  10. Jika B = ⎟⎟ ⎠ ⎞

  ⎠ ⎞

  3

  2

  1

  2

  3

  1

  2 X = ^{1 −} A . C

  =

  ⎟ ⎟ ⎠ ⎞

  ⎜ ⎜ ⎝ ⎛

  − −

  2

  1

  2

  1

  6 Jawabannya adalah C UMPTN1990

  2 .

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  1

  2

  3

  4 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  4

  5

  5

  9 B.

  ⎝ ⎛

  ⎜ ⎜ ⎝ ⎛

  1 −

  13

  9

  5 Jawabannya adalah A bukti: AB ^{1 −} =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  3

  4

  1

  2 ,

  B ^{1 −} =

  6

  5

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  ⎜⎜ ⎝ ⎛

  − −

  1

  3

  2

  5 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  1

  3

  2

  5

  23

  ⎟⎟ ⎠ ⎞

  13

  −

  9

  3

  5 D.

  ⎠ ⎞

  ⎝ ⎛

  10

  2

  5

  13 Jawab: A.B ¹

  −

  = C A = C . (B ^{1 −} ) ^{1 −} (B ¹

  −

  ) ¹

  = B ^{1 1 −}

  1 =

  − x

  = B maka A = C .B =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  3

  4

  1

  2 .

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  5

  3

  2

  − −

  ⎟ ⎟ ⎠ ⎞

  B – A = C ^t ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  2

  1

  ⎜⎜ ⎝ ⎛

  1 X = ⎟⎟ ⎠ ⎞

  2

  3

  4

  9. Matriks X berordo 2 x 2 yang memenuhi persamaan ⎟⎟ ⎠ ⎞

  4 adalah… A.

  7 y – 4 = 1 y = 5 x + y – 2 = 7 x + 5 – 2 = 7 x = 7 – 5 +2 x = 4 x . y = 4 . 5 = 20 jawabannya dalah C EBTANAS1992

  3

  2

  1

  ⎜⎜ ⎝ ⎛

  = ⎟⎟ ⎠ ⎞

  y y x

  3

  ⎟⎟ ⎠ ⎞

  3

  ⎜⎜ ⎝ ⎛

  ⎜⎜ ⎝ ⎛

  − −

  4

  5

  5

  6 E.

  ⎟⎟ ⎠ ⎞

  − −

  ⎜⎜ ⎝ ⎛

  5

  4

  6

  5 B.

  1

  1

  4 =

  2

  2

  4 C.

  4 =

  1 − .

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  1

  3

  2

  2

  6

  1 −

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  1

  3

  2

  4

  ⎜⎜ ⎝ ⎛ +

  4

  ⎟⎟ ⎠ ⎞

  − − +

  ⎜⎜ ⎝ ⎛

  7 ⎟⎟ ⎠ ⎞

  3

  2

  1

  ⎜⎜ ⎝ ⎛

  2 =

  y y x

  1

  1

  4

  −

  ⎜⎜ ⎝ ⎛

  2

  3

  ⎟⎟ ⎠ ⎞

2 Jawab:

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  2

  3

  4 Misal A = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  4

  3

  2

  1 dan C = ⎟⎟ ⎠ ⎞

  1

  ⎜⎜ ⎝ ⎛

  2

  3

  4 Maka X = ^{1 −}

  A . C _{1 −} A = − bc ad

  1 .

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  a c b d ^{1 −} A =

  1

  1 X = ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  2

  1

  2 D.

  ⎟ ⎟ ⎠ ⎞

  ⎜ ⎜ ⎝ ⎛

  − −

  2

  1

  1

  1

  2

  1

  Teori: Jika A.B = C maka 1. A = C . ^{1 −}

  B

  2. B = ^{1 −}

  A . C

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  4

  3

  1

  ⎟⎟ ⎠ ⎞

  65

  5 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − + − − + −

  23

  26

  69

  9

  5 3 . 9 )

  10

  27

  25 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  3

  4

  1

  5 (

  9 2 .

  ⎜⎜ ⎝ ⎛

  1

  23

  13

  9

  5 .

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − −

  3

  1 (

  2

  5 =

  ⎟⎟ ⎠ ⎞

  ⎜⎜ ⎝ ⎛

  − + + − − + + −

  ) 1 ( 23 2 .

  13 3 . 23 ) 5 .(

  13 )

  2 Æ terbukti

  2 E. Transpose Matriks :

  13. A = A . A _{3 2} A = A . A _{4 3}

  a b a c

  ⎛ ⎞ t ⎛ ⎞ A = A . A

  A

  Jika A = , maka = ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟

c d b d .

  ⎝ ⎠ ⎝ ⎠ _t .

  .

  A didapat dari mengubah kedudukan baris menjadi kolom

  dari matriks A

  F. Persamaan Matriks : _{n n} − ¹ A = A. A

  Jika A.B = C maka

  − ¹ B 1. A = C .

  − ¹

  2. B = A . C ( urutan huruf diperhatikan !!)

G. Sifat-sifat Operasi Matriks :

  1. A + B = B + A (sifat komutatif)

  2. A . B ≠

  B. A

  3. A. (B. C) = (A . B) . C (sifat asosiatif) 4. (A + B) + C = A + ( B + C )

  ⎛ ⎞

  5. A + O = O + A ; O = ⎜⎜ ⎟⎟ ⎝ ⎠

  6. A + (-A) = 0

  7. A – B = A + (-B)

  − ^{1 − 1}

  8. ( A ) = A _{t t} 9. ( A ) = A _{1 1 1}

  − − −

  10. ( A . B ) = B . A _{t t t} 11. ( A . B ) = B . A

  1 ⎛ ⎞

  − ¹

  12. A . A = I = ⎜⎜ ⎟⎟

  1 ⎝ ⎠

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