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# 19. soal soal matriks

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### 19. soal soal matriks

1. 1. www.matematika-sma.com - 1 19. SOAL-SOAL MATRIKS EBTANAS1998 1. Diketahui matriks A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 72 14 ; B = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 72 14 dan C= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 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 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 72 14 +3 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 72 14 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 14 8 b a ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 72 14 + 3 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 72 14 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 72 14 + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 216 312 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+− +−− 21762 31124 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 144 28 Didapat a = 2 dan b = 4 Jawabannya adalah A EBTANAS2000 2. Diketahui matrik A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− p4 24 B= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 43 81 , dan C= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 814 242 Jika AB=C, nilai p=… A. -6 B. - 3 10 C. 3 1 D. 3 10 E. 6 Jawab : A.B = C ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− p4 24 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 43 81 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 814 242 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− p4 24 ’ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 43 81 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −++− −−+−−+−− )4.(8.43.)1.(4 )4.(28.43).2()1.(4 pp = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+− −− pp 432.34 242 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 814 242 -4 + 3p = 14 32 – 4p = 8 3p = 18 32 – 8 = 4p = 6 24 = 4p p = 6 jawabannya adalah E UAN2004 3. Diketahui matriks A= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 23 58 , B = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 23 2x dan C = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + 43 539 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 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 23 58 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 23 2x = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 23 58 + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + 43 539 y ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 23 58 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 23 2x = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+−+ −+−+ 2).2(2.33).2(.3 2).5(2.83).5(.8 x x = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 26.3 615.8 x x ….(1)
2. 2. www.matematika-sma.com - 2 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 23 58 + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + 43 539 y = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +−+ ++−+ 4233 53598 y = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 26 317 y …(2) (1) = (2) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 26.3 615.8 x x = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 26 317 y 8x-15 = 17 3y = 6 8x = 32 y = 2 x = 4 x + y = 4 + 2 = 6 jawabannya adalah D EBTANAS2000 4. Diketahui A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 21 32 , B = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 104 126 Dan A 2 = x.A + y.B, nilai xy=… 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 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 21 32 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 21 32 = x. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 21 32 + y. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 104 126 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 21 32 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 21 32 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −−+−−−+− −+−+ )2).(2(3.1)1).(2(2.1 )2.(33.2)1.(32.2 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 10 01 …(1) x. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 21 32 + y. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 104 126 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− xx xx 2 32 + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− yy yy 104 126 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −−−− ++ yxyx yxyx 1024 12362 …(2) (1) = (2) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 10 01 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −−−− ++ yxyx yxyx 1024 12362 2x + 6y = 1 x 3 ⇒ 6x + 18y = 3 3x+12y = 0 x 2 ⇒ 6x+ 24 y = 0 - 0 - 6y = 3 y = - 2 1 6x+ 24 y = 0 6x = -24y 6x = -24 . (- 2 1 ) 6x = 12 x = 2 x. y = 2. - 2 1 = - 1 jawabannya adalah B UAN2004 5. Diketahui matriks S = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 31 02 dan M = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 30 21 . Jika fungsi f(S,M) = S 2 -M 2 matriks f(S+M, S-M) adalah… A. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 404 204 D. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− − 404 204 B. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 304 204 E. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 364 84 C. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 384 84 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 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 31 02 + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 30 21 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 01 23
3. 3. www.matematika-sma.com - 3 (S+M) 2 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 01 23 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 01 23 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 23 67 S – M = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 31 02 - ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 30 21 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −−−− −− )3(301 2012 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 61 21 (S-M) 2 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 61 21 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 61 21 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 387 143 (S+M) 2 - (S-M) 2 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 23 67 - ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 387 143 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 404 204 Jawabannya adalah A EBTANAS1997 6. Diketahui A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 86 3 x adalah matriks singular. Nilai x = …. A. -5 B. -4 C. -3 D. 3 E. 4 Jawab: teori: Jika A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ dc ba Maka det(A) = |A| = ad – bc jika det(A) = 0 maka matriks A disebut matriks singular A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 86 3 x Det(A) = ad – bc = 3.8 – (-x).6 = 24 + 6x =0 6x = -24 x = -4 jawabannya adalah B UAN2006 7. Diketahui matriks A= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 15 43 dan B = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 72 21 jika M = A + B, maka invers M adalah M 1− = …. A. ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − 4 2 1 3 11 C. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 87 22 E. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 87 22 B. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− −− 87 22 D. ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − 1 2 1 3 14 Jawab: M = A + B = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 15 43 + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 72 21 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 87 22 M 1− = bcad − 1 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − ac bd = 7.28.2 1 − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 27 28 = 2 1 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 27 28 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − 1 2 1 3 14 jawabannya adalah D UAN2007 8. Diketahui matriks A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 41 12 ; B = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + y yx 3 2 dan C = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 13 27 apabila B – A = Ct dan Ct = transpose matriks C, maka nilai x. y = … A. 10. B. 15 C. 20 D. 25 E. 30 jawab: teori : Jika A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ dc ba , maka =t A ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ db ca C = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 13 27 Ct = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 12 37
4. 4. www.matematika-sma.com - 4 B – A = Ct ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + y yx 3 2 - ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 41 12 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 12 37 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − −+ 42 32 y yx = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 12 37 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 9. Matriks X berordo 2 x 2 yang memenuhi persamaan ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 43 21 X = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 12 34 adalah… A. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 10 14 C. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 45 56 E. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 54 65 B. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 01 12 D. ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − 2 1 1 2 1 12 Jawab: Teori: Jika A.B = C maka 1. A = C . 1− B 2. B = 1− A . C ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 43 21 X = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 12 34 Misal A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 43 21 dan C = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 12 34 Maka X = 1− A . C 1− A = bcad − 1 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − ac bd 1− A = 64 1 − . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 13 24 = 2 1 − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 13 24 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − 2 1 2 3 12 X = 1− A . C = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − 2 1 2 3 12 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 12 34 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− 45 56 Jawabannya adalah C UMPTN1990 10. Jika B = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 53 21 dan AB 1− = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 34 12 , maka A =… A. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2313 95 C. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 239 53 E. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 312 59 B. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 139 35 D. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 102 513 Jawab: A.B 1− = C A = C . (B 1− ) 1− (B 1− ) 1− = B 11 −− x = B maka A = C .B = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 34 12 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 53 21 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2313 95 Jawabannya adalah A bukti: AB 1− = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 34 12 , B 1− = 65 1 − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 13 25 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 13 25
5. 5. www.matematika-sma.com - 5 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2313 95 . ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − 13 25 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −++− −++− )1(232.133.23)5.(13 )1(92.53.9)5(5 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+− −+− 23266965 9102725 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 34 12 terbukti