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Ilhamsyah Ibnu Hidayat Menyelesaikan Masalah Berkaitan dengan Konsep Matriks

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Page 1: Soal Matriks

Ilhamsyah Ibnu Hidayat

Menyelesaikan Masalah Berkaitan dengan Konsep Matriks

XI AdministrasiPerkantoran

Page 2: Soal Matriks

Matriks

Pilihan Ganda

1. Jika A = [1 02 3] dan I matriks satuan ordo dua, maka A2 - 2A + I =

a.[0 00 0] c.[0 0

4 4] c. [1 02 3]

b.[0 12 3 ] d.[4 4

0 0]2. Diketahui matriks A = [1 2

4 3] dan I = [1 00 1]. Tentukan nilai x supaya

matriks A - xI merupakan matriks singular !

a. −15

b. 29

c. 12

23

e. 5

−2

3. Tentukan invers matriks A = [ 2 −3−2 4 ]a. [2 4

6 8 ] c.

[2 32

1 1 ] e. [1 23 4]b. [ 2 −3

−2 4 ] d.

[−3 42 −2]

4. Jika A = [2 51 3] dan B = [5 4

1 1] maka tentukan determinan (AB)-1 !

a.1 b. 2 c. 3 d. 4 e. 5

5. Tentukan matriks P jika [3 41 2] P = [1 2

3 4]!a. [−5 −6

4 5 ] c. [5 45 4] e.[−5 −6

5 4 ]b.[−6 −5

4 5 ] d. [−6 −55 4 ]

6. Diketahui A = [2 10 −1] dan B = [−1 1

0 2]. Tentukan nilai A -2B!

a. [4 −10 −5] c. [−5 4

−1 0] e. [0 −54 −1]d.

[ 4 0−5 −1] d. [ 4 −1

−5 0 ]7. Diketahui A = [ 2 −3 1

−4 0 4 ] dan B = [ 1 −5−2 43 6 ]. Tentukan -2AB !

Page 3: Soal Matriks

a. [−22 32−88 −16] c. [−88 −22

−16 32 ] e. [−22 −8832 −16 ]

b. [−88 −16−22 32 ] d. [−22 32

−16 −88]8. Diketahui A = [2 1

3 2], B = [4 32 3] dan C = [5 1

4 2]. Tentukan AB – C!

a. [ 5 812 13] c. [12 8

13 5 ] e. [13 812 5 ]

b. [12 135 8 ] d. [5 12

8 13 ]

9. Diketahui A = [ x+ y xy x− y ] dan B = [ 1

−12x

−2 y 3 ]. Jika A1 menyatakan

matriks transpose dari A, maka tentukan x jika A1 = B!a. 1 b. 2 c. 3 d. 4 e. 5

10. Diketahui [5 a 3b 2 c ]. Tentukana + b + c !

a. 14 b. 15 c. 16 d. 17 e. 18

11. Diketahui A = [ a 42b 3c ] dan B = [2c−3b 2a+1

a b+7 ]. Jika A = 2B1, maka

tentukan c !a. 2 b. 8 c. 4 d. 6 e. 7

12. Diketahui [ x −2−4 y ] + 2[−1 3

4 x ] = [ y 44 10]. Tentukan x !

a. 4 b. 6 c. 2 d. 10 e. 8

13. Diketahui [ x log y 2 log z1 3 log y] = [4 log z 2

112 ]. Tentukan x !

a. 2 b. 2√2 c. √3 d. 4 e. 4, 5

14. Diketahui A = [2x −53 y ], B = [ y 2

2 4] dan C = [8 −35 2 x ]. Tentukan nilai x

+ y yang memenuhi A + B = Ca. 5 b. 7 c. 9 d. 11 e. 3

Page 4: Soal Matriks

15. Diketahui A = [1 a+bb c ], B = [a−1 0

−c d ] dan C = [1 01 1]. Jika A + B1 = C2,

maka tentukan da. 0 b. -2 c. 7 d. 4 e. -5

16. Diketahui A = [−4 −24 p ], B = [−1 8

3 −4] dan C = [−2 −2414 8 ]. Jika AB = C,

maka tentukan pa. 2 b. 3 c. 4 d. 5 e. 6

17. Diketahui [−1 d−b 3 ] + [ 4 −5

−3 b ] = [ 2 −1−4 3 ] [2c 1

c a+1]. Tentukan a !

a. 0 b. 8 c. 4 d. 7 e. 2

18. Jika A = [1 42 3] dan I = [1 0

0 1] memenuhi persamaan A2 = pA + qI,

maka p – q = ...a. 5 b. 7 c. 9 d. -5 e. -1

19. Jika α , β, dan γ adalah sudut-sudut segitiga ABC dan

[ sinα cos αcos β sinβ ] [cos β −sin β

sin β cos β ] = [sin γ cos12γ

1 0 ], maka tentukan γ !

a. 120o b. 2700 c. 360o d. 90o e. 45o

20. Hasil kali matriks (BA) (B + A-1) B-1- = ...a. BA + I b. IA + B c. BI + A d. A + IB e. I

Essay

21. Tentukan nilai x yang memenuhi persamaan [ x x2 x ] = [−2 −2

2 −2] !

22. Diketahui A = [2 13 4], B = [−1 2

5 6 ] dan C = [a −12 9 ]. Jika determinan 2A

– B + 3C adalah 10, maka tentukan nilai a !

23. Diketahui A = [5+x −x5 3 x ] dan B = [9 −x

7 4 ]. Jika |A| = |B|, maka

tentukan x !

Page 5: Soal Matriks

24. Tentukan nilai determinan matriks [ 0 2 3−2 0 4−3 −4 0 ] !

25. Diketahui matriks A = [1 23 4]. Jika AB = [1 0

0 1], maka tentukan matriks

B !

Jawaban Pilihan Ganda

1. (C) Jawab :

A2 - 2A + I = [1 02 3] [1 0

2 3] – 2[1 02 3] + [1 0

0 1] = [0 04 4]

2. (A) Jawab :

A – xI = [1 24 3] - [ x 0

0 x ] = [1−x 24 3−x ]

Matriks singular syaratnya berdeterminan 0, sehingga :

|1−x 24 3−x| = 0 ↔ ( 1 – x ) (3 – x ) – 8 = 0 ↔ x = -1 atau x = 5

3. (C) Jawab :

A-1 = 1

2.4−(−2 )(−3) [4 32 2] = [2 3

21 1 ]

4. (A) Jawab :

Page 6: Soal Matriks

A =[2 51 3]→|A| = 6 – 5 = 1

B = [5 41 1]→|B| = 5 – 4 = 1

|(AB)-1| = 1

|AB| = 1

|A||B| = 11.1

= 1

5. (D) Jawab

P = [3 41 2]-1 [2 1

4 3] = 16−4 [ 2 −4

−1 3 ][2 14 3] = [−6 −5

5 4 ]6. (A) Jawab

A – 2B = [2 10 −1] - [−2 2

0 4] = [4 −10 −5]

7. (D) Jawab

-2AB = [−4 6 −28 0 −8] [ 1 −5

−2 43 6 ] = [−22 32

−16 −88]8. (A) Jawab

AB – C = [2 13 4] [4 3

2 3] - [5 14 2] = [ 5 8

12 13]9. (B) Jawab

A1 = B → [ x+ y yx x− y ] = [ 1

−12x

−2 y 3 ]x + y = 1x – y = 3 x = 2

10. (A) Jawaba = 2 → b = 2a = 4 → c = ab = 8a + b + c = 14

11. (B) Jawab

A = 2B1 → [ a 42b 3c ] = 2[2c−3b a

2a+1 b+7][ a 42b 3c ] = [4 c−6b 2a

4 a+2 2b+14]2a = 4 ↔ a = 22b = 4.2 + 2 ↔ b = 53c = 2.5 +14 ↔ c = 8

Page 7: Soal Matriks

12. (A) Jawab

[ x−2 44 y+2 x ] = [ y 4

4 10]x – y = 22x + y = 10 → x = 4

13. (C) Jawab2log z = 2 ↔ z = 4

3log y = 12 ↔ y = √3

3log y = 4log z → xlog √3 = 4log 4 ↔ x = √3

14. (A) Jawab

A + B = C → [2x+ y −35 y+4 ] =[8 −3

5 2 x ]2x + y = 8y + 4 = 2x x = 3 dan y = 2x + y = 5

15. (B) JawabA + B1 = C2

[1 a+bb c ] + [a−1 −c

0 d ] = [1 01 1] [1 0

1 1][a a+b−ca c+d ] = [1 0

2 1]a = 1 dan b = 2a + b – c = 0 → c = 1 + 2 = 3c + d = 1 → d = 1 – 3 = -2

16. (E) Jawab

AB = C → [−4 −24 p ] [−1 8

3 −4] = [−2 −2414 8 ]

[ −2 −243 p−4 32−4 p] = [−2 −24

14 8 ]3p – 4 = 14 ↔ p = 6

17. (E) Jawab

[ 3 d−5−b−3 3+b ] = [ 3c −a+1

−5c 3a−1]3 = 3c ↔ c =1-b – 3 = -5c → b = 5.1 -3 = 23 + b = 3a – 1 → 3 + 2 = 3a – 1 ↔ a = 2

Page 8: Soal Matriks

18. (E) Jawab

A2 = pA + qI → [1 42 3] [1 4

2 3 ] = [ p 4 p2 p 3 p] = [q 0

0 0][9 168 17 ] = [ p+q 4 p

2 p 3 p+q]8 = 2p ↔ p = 49 = p + q → q = 5P – q = 4 – 5 = -1

19. (A) Jawab

[sinα cos β+cos α sin β cos α cos β−sinα sin βcos ² β+sin ² β 0 ]= [sin γ cos

12γ

1 0 ][sin(α+β) cos (α+β )

1 0 ] = [sin γ cos12γ

1 0 ]cos (α+β) = cos

12γ

cos (180o – γ) = cos 12γ

- cos γ = cos 12γ

- (2 cos2 12γ-1) = cos

12γ

(2 cos 12γ-1) (cos

12γ+ 1) = 0

cos 12γ =

12 → γ

= 120o

cos 12γ = -1 → γ = 360o

20. (A) Jawab(BA) (B+A-1) B-1 = (BA) (BB-1 + A-1B-1)= BA) (I + A-1B-10 = BA + BAA-1B-1 = BA + IJawaban Essay

21. X2 – 2x = 4 + 4 ↔ (x – 4) (x + 2) = 0 ↔ x = 4 atau x = -2

Page 9: Soal Matriks

22. |2A – B + 3C| = [3 A+5 −37 11 ] = 10

(3a + 5) . 11 + 21 = 10 ↔ a = -2

23. (5 + x) (3x) – 5x = 36 + 7x ↔ (x + 4) (x – 3) = 0x = -4 atau x = 3

24. ( 0 2 3−2 0 4−3 −4 0|

0 2−2 0−3 −4) = 0 – 24 + 24 – 0 – 0 – 0 = 0

25. AB = I → B = A-1 = 14−6 [ 4 −2

−3 1 ] = [−2 112

−12 ]

Page 10: Soal Matriks

Sumber

http://www.slideshare.net/hancee/matrikssoal-jawab

Membuat soal dan penjabaran sendiri