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Given A=(21​−13​), B=(5p​m4​), and C=(n4​6t​), find the value of m,n,p,andt when  b.    3A−2B=4C

Pertanyaan

Given begin mathsize 14px style A equals open parentheses table row 2 cell negative 1 end cell row 1 3 end table close parentheses end stylebegin mathsize 14px style B equals open parentheses table row 5 m row p 4 end table close parentheses end style, and begin mathsize 14px style C equals open parentheses table row n 6 row 4 t end table close parentheses end style, find the value of begin mathsize 14px style m comma space n comma space p comma space and space t end style when 

b.    begin mathsize 14px style 3 A minus 2 B equals 4 C end style 

Pembahasan Soal:

Selisih dua matriks adalah sebuah matriks baru yang berordo sama, yaitu elemen-elemennya merupakan hasil pegurangan elemen-elemen matriks A dan B.

open parentheses table row a b row c d end table close parentheses minus open parentheses table row k l row m n end table close parentheses equals open parentheses table row cell a minus k end cell cell b minus l end cell row cell c minus m end cell cell d minus n end cell end table close parentheses  

Perkalian matriks M dengan skalar k dituliskan dengan k times M, adalah sebuah matriks yang dibentuk dengan mengalikan masing-masing elemen dari matriks M dengan k.

k. M equals k times open parentheses table row a b row c d end table close parentheses equals open parentheses table row cell k times a end cell cell k times b end cell row cell k times c end cell cell k times d end cell end table close parentheses 

Berdasarkan aturan operasi matriks di atas, didapat perhitungan berikut ini.

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 A minus 2 B end cell equals cell 4 C end cell row cell 3 open parentheses table row 2 cell negative 1 end cell row 1 3 end table close parentheses minus 2 open parentheses table row 5 m row p 4 end table close parentheses end cell equals cell 4 open parentheses table row n 6 row 4 t end table close parentheses end cell row cell open parentheses table row 6 cell negative 3 end cell row 3 9 end table close parentheses minus open parentheses table row 10 cell 2 m end cell row cell 2 p end cell 8 end table close parentheses end cell equals cell open parentheses table row cell 4 n end cell 24 row 16 cell 4 t end cell end table close parentheses end cell row cell open parentheses table row cell negative 4 end cell cell negative 3 minus 2 m end cell row cell 3 minus 2 p end cell 1 end table close parentheses end cell equals cell open parentheses table row cell 4 n end cell 24 row 16 cell 4 t end cell end table close parentheses end cell end table    

Dari kesamaan matriks tersebut, diperoleh:

  • elemen baris 1 dan kolom 1: 

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 n end cell equals cell negative 4 end cell row cell fraction numerator 4 n over denominator 4 end fraction end cell equals cell fraction numerator negative 4 over denominator 4 end fraction end cell row cell therefore space n end cell equals cell negative 1 end cell end table 

  • elemen baris 1 dan kolom 2: 

table attributes columnalign right center left columnspacing 0px end attributes row cell negative 3 minus 2 m end cell equals 24 row cell negative 3 minus 2 m plus 3 end cell equals cell 24 plus 3 end cell row cell fraction numerator negative 2 m over denominator negative 2 end fraction end cell equals cell fraction numerator 27 over denominator negative 2 end fraction end cell row cell therefore space m end cell equals cell negative 27 over 2 end cell end table   

  • elemen baris 2 dan kolom 1: 

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 minus 2 p end cell equals 16 row cell 3 minus 2 p minus 3 end cell equals cell 16 minus 3 end cell row cell fraction numerator negative 2 p over denominator negative 2 end fraction end cell equals cell fraction numerator 13 over denominator negative 2 end fraction end cell row cell therefore space p end cell equals cell negative 13 over 2 end cell end table   

  • elemen baris 2 dan kolom 2: t equals 10 

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 t end cell equals 1 row cell fraction numerator 4 t over denominator 4 end fraction end cell equals cell 1 fourth end cell row cell therefore space t end cell equals cell 1 fourth end cell end table

Jadi, nilai m comma space n comma space p comma dan t pada saat begin mathsize 14px style 3 A minus 2 B equals 4 C end style adalah n equals negative 1m equals negative 27 over 2p equals negative 13 over 2, dan t equals 1 fourth.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

F. Nur

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Diberikan A=(ac​bd​), B=(−ac​b−d​), dan C=(ac​−b−d​). Tentukan hasil dari setiap bentuk aljabar matriks berikut. c.    2(A−C)2+2(B−C)2−(A+B−2C)2

Pembahasan Soal:

Diketahui:

begin mathsize 14px style A equals open parentheses table row a b row c d end table close parentheses end style
begin mathsize 14px style B equals open parentheses table row cell negative a end cell b row c cell negative d end cell end table close parentheses end style
begin mathsize 14px style C equals open parentheses table row a cell negative b end cell row c cell negative d end cell end table close parentheses end style

Ditanya:

begin mathsize 14px style 2 left parenthesis A minus C right parenthesis squared plus 2 left parenthesis B minus C right parenthesis squared minus left parenthesis A plus B minus 2 C right parenthesis squared end style

Perkalian matriks merupakan suatu nilai matriks yang bisa dihasilkan dengan cara tiap baris dikalikan dengan setiap kolom yang jumlah pada barisnya sama.

Hasil dari 2 open parentheses A minus C close parentheses squared:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 open parentheses A minus C close parentheses squared end cell equals cell 2 open parentheses open parentheses table row a b row c d end table close parentheses minus open parentheses table row a cell negative b end cell row c cell negative d end cell end table close parentheses close parentheses squared end cell row blank equals cell 2 open parentheses table row cell a minus a end cell cell b minus left parenthesis negative b right parenthesis end cell row cell c minus c end cell cell d minus open parentheses negative d close parentheses end cell end table close parentheses squared end cell row blank equals cell 2 open parentheses table row 0 cell 2 b end cell row 0 cell 2 d end cell end table close parentheses squared end cell row blank equals cell 2 open parentheses table row 0 cell 2 b end cell row 0 cell 2 d end cell end table close parentheses open parentheses table row 0 cell 2 b end cell row 0 cell 2 d end cell end table close parentheses end cell row blank equals cell 2 open parentheses table row cell 0 plus 0 end cell cell 0 plus 4 b d end cell row cell 0 plus 0 end cell cell 0 plus 4 d squared end cell end table close parentheses end cell row blank equals cell 2 open parentheses table row 0 cell 4 b d end cell row 0 cell 4 d squared end cell end table close parentheses end cell row blank equals cell open parentheses table row 0 cell 8 b d end cell row 0 cell 8 d squared end cell end table close parentheses end cell end table 

Hasil dari 2 open parentheses B minus C close parentheses squared:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 open parentheses B minus C close parentheses squared end cell equals cell 2 open parentheses open parentheses table row cell negative a end cell b row c cell negative d end cell end table close parentheses minus open parentheses table row a cell negative b end cell row c cell negative d end cell end table close parentheses close parentheses squared end cell row blank equals cell 2 open parentheses table row cell negative a minus a end cell cell b minus open parentheses negative b close parentheses end cell row cell c minus c end cell cell negative d minus open parentheses negative d close parentheses end cell end table close parentheses squared end cell row blank equals cell 2 open parentheses table row cell negative 2 a end cell cell 2 b end cell row 0 0 end table close parentheses squared end cell row blank equals cell 2 open parentheses table row cell negative 2 a end cell cell 2 b end cell row 0 0 end table close parentheses open parentheses table row cell negative 2 a end cell cell 2 b end cell row 0 0 end table close parentheses end cell row blank equals cell 2 open parentheses table row cell 4 a squared plus 0 end cell cell negative 4 a b plus 0 end cell row 0 0 end table close parentheses end cell row blank equals cell 2 open parentheses table row cell 4 a squared end cell cell negative 4 a b end cell row 0 0 end table close parentheses end cell row blank equals cell open parentheses table row cell 8 a squared end cell cell negative 8 a b end cell row 0 0 end table close parentheses end cell end table 

Hasil dari open parentheses A plus B minus 2 C close parentheses squared:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses A plus B minus 2 C close parentheses squared end cell equals cell open parentheses open parentheses table row a b row c d end table close parentheses plus open parentheses table row cell negative a end cell b row c cell negative d end cell end table close parentheses minus 2 open parentheses table row a cell negative b end cell row c cell negative d end cell end table close parentheses close parentheses squared end cell row blank equals cell open parentheses open parentheses table row a b row c d end table close parentheses plus open parentheses table row cell negative a end cell b row c cell negative d end cell end table close parentheses minus open parentheses table row cell 2 a end cell cell negative 2 b end cell row cell 2 c end cell cell negative 2 d end cell end table close parentheses close parentheses squared end cell row blank equals cell open parentheses table row cell a minus a minus 2 a end cell cell b plus b minus open parentheses negative 2 b close parentheses end cell row cell c plus c minus 2 c end cell cell d minus d minus open parentheses negative 2 d close parentheses end cell end table close parentheses squared end cell row blank equals cell open parentheses table row cell negative 2 a end cell cell 4 b end cell row 0 cell 2 d end cell end table close parentheses squared end cell row blank equals cell open parentheses table row cell negative 2 a end cell cell 4 b end cell row 0 cell 2 d end cell end table close parentheses open parentheses table row cell negative 2 a end cell cell 4 b end cell row 0 cell 2 d end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 4 a squared plus 0 end cell cell negative 8 a b plus 8 b d end cell row cell 0 plus 0 end cell cell 0 plus 4 d squared end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 4 a squared end cell cell negative 8 a b plus 8 b d end cell row 0 cell 4 d squared end cell end table close parentheses end cell end table

Maka, hasil dari begin mathsize 14px style 2 left parenthesis A minus C right parenthesis squared plus 2 left parenthesis B minus C right parenthesis squared minus left parenthesis A plus B minus 2 C right parenthesis squared end style sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 left parenthesis A minus C right parenthesis squared plus 2 left parenthesis B minus C right parenthesis squared minus left parenthesis A plus B minus 2 C right parenthesis squared end cell row blank equals cell open parentheses table row 0 cell 8 b d end cell row 0 cell 8 d squared end cell end table close parentheses plus open parentheses table row cell 8 a squared end cell cell negative 8 a b end cell row 0 0 end table close parentheses minus open parentheses table row cell 4 a squared end cell cell negative 8 a b plus 8 b d end cell row 0 cell 4 d squared end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 8 a squared minus 4 a squared end cell cell 8 b d minus 8 a b plus 8 a b minus 8 b d end cell row 0 cell 8 d squared minus 4 d squared end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 4 a squared end cell 0 row 0 cell 4 d squared end cell end table close parentheses end cell end table

Jadi,  hasil dari begin mathsize 14px style 2 left parenthesis A minus C right parenthesis squared plus 2 left parenthesis B minus C right parenthesis squared minus left parenthesis A plus B minus 2 C right parenthesis squared end style adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell 4 a squared end cell 0 row 0 cell 4 d squared end cell end table close parentheses end cell end table.

0

Roboguru

Jika matriks-matriks A dan B berordo 2×2 mempunyai sifat A⋅B=B⋅A, tunjukkanlah: b.    (2A−3B)(A+2B)=2A2+4AB−3BA−6B2

Pembahasan Soal:

Ingat kembali:

Jika matriks persegi begin mathsize 14px style A end styleB, dan C berordo m cross times m dan k dan hbilangan real, maka berlaku:

  • sifat distributif perkalian open parentheses A minus B close parentheses C equals open parentheses A times C close parentheses minus open parentheses B times C close parentheses
  • h open parentheses k times A close parentheses equals open parentheses h times k close parentheses A
  • A times A equals A squared
  • sifat komutatif penjumlahan A plus B equals B plus A
  • A minus B equals A plus open parentheses negative B close parentheses

Dengan demikian, 

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 2 A minus 3 B close parentheses open parentheses A plus 2 B close parentheses end cell equals cell open parentheses 2 A minus 3 B close parentheses A plus open parentheses 2 A minus 3 B close parentheses 2 B end cell row blank equals cell open parentheses 2 A times A minus 3 B times A close parentheses plus open parentheses 2 A times 2 B minus 3 B times 2 B close parentheses end cell row blank equals cell open parentheses 2 A times A minus 3 B times A close parentheses plus open parentheses open parentheses 2 times 2 close parentheses open parentheses A times B close parentheses minus open parentheses 3 times 2 close parentheses open parentheses B times B close parentheses close parentheses end cell row blank equals cell open parentheses 2 A times A minus 3 B times A close parentheses plus open parentheses 4 open parentheses A times B close parentheses minus 6 open parentheses B times B close parentheses close parentheses end cell row blank equals cell 2 A squared minus 3 B A plus 4 A B minus 6 B squared end cell row blank equals cell 2 A squared plus open parentheses negative 3 B A close parentheses plus 4 A B minus 6 B squared end cell row blank equals cell 2 A squared plus 4 A B plus left parenthesis negative 3 B A right parenthesis minus 6 B squared end cell row blank equals cell 2 A squared plus 4 A B minus 3 B A minus 6 B squared end cell end table end style

Jadi, open parentheses 2 A minus 3 B close parentheses open parentheses A plus 2 B close parentheses equals 2 A squared plus 4 A B minus 3 B A minus 6 B squared dengan A times B not equal to B times A.space 

0

Roboguru

Diketahui A,B,C,danX matriks berordo sama. Selesaikan X dalam A,B,danC dari setiap persamaan berikut. b.    A−{B−(C+X)}=2X−{C−(2A+B)}

Pembahasan Soal:

Dua buah matriks hanya dapat dijumlahkan apabila kedua matriks itu berordo sama.

Diketahui pers.  begin mathsize 14px style A minus open curly brackets B minus left parenthesis C plus X right parenthesis close curly brackets equals 2 X minus open curly brackets C minus left parenthesis 2 A plus B right parenthesis close curly brackets end style,

sehingga matriks begin mathsize 14px style X end style dalam begin mathsize 14px style A comma space B comma space dan space C end style:

table attributes columnalign right center left columnspacing 0px end attributes row cell A minus open curly brackets B minus left parenthesis C plus X right parenthesis close curly brackets end cell equals cell 2 X minus open curly brackets C minus left parenthesis 2 A plus B right parenthesis close curly brackets end cell row cell A minus open parentheses B minus C minus X close parentheses end cell equals cell 2 X minus open parentheses C minus 2 A minus B close parentheses end cell row cell A minus B plus C plus X minus X end cell equals cell 2 X minus C plus 2 A plus B minus X end cell row cell A minus B plus C end cell equals cell X minus C plus 2 A plus B end cell row cell A minus B plus C minus B end cell equals cell X minus C plus 2 A plus B minus B end cell row cell A minus 2 B plus C end cell equals cell X minus C plus 2 A end cell row cell A minus 2 B plus C minus 2 A end cell equals cell X minus C plus 2 A minus 2 A end cell row cell negative A minus 2 B plus C end cell equals cell X minus C end cell row cell negative A minus 2 B plus C plus C end cell equals cell X minus C plus C end cell row cell negative A minus 2 B plus 2 C end cell equals X end table  

Jadi, matriks begin mathsize 14px style X end style dalam begin mathsize 14px style A comma space B comma space dan space C end style adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank X end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank A end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank B end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table.

0

Roboguru

Diketahui matriks A=[24​58​],B=[73​9−2​] dan C=[48​11​]. Hasil operasi A+2B−C adalah...

Pembahasan Soal:

Sesuai dengan sifat penjumlahan dan pengurangan matriks, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell A plus 2 B minus C end cell equals cell open square brackets table row 2 5 row 4 8 end table close square brackets plus 2 open square brackets table row 7 9 row 3 cell negative 2 end cell end table close square brackets minus open square brackets table row 4 1 row 8 1 end table close square brackets end cell row blank equals cell open square brackets table row 2 5 row 4 8 end table close square brackets plus open square brackets table row 14 18 row 6 cell negative 4 end cell end table close square brackets minus open square brackets table row 4 1 row 8 1 end table close square brackets end cell row blank equals cell open square brackets table row cell open parentheses 2 plus 14 minus 4 close parentheses end cell cell open parentheses 5 plus 18 minus 1 close parentheses end cell row cell open parentheses 4 plus 6 minus 8 close parentheses end cell cell open parentheses 8 plus open parentheses negative 4 close parentheses minus 1 close parentheses end cell end table close square brackets end cell row blank equals cell open square brackets table row 12 22 row 2 3 end table close square brackets end cell end table 

Oleh karena itu, jawaban yang benar adalah B.

0

Roboguru

Diketahui K=(41​02​), L=(13​−10​), dan M=(31​12​), maka K+L−2M=....

Pembahasan Soal:

Pada perkalian matriks skalar, elemen-elemen matriks dikalikan dengan bilangan skalarnya, sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 M end cell equals cell 2 open parentheses table row 3 1 row 1 2 end table close parentheses end cell row blank equals cell open parentheses table row 6 2 row 2 4 end table close parentheses end cell end table

Ingat bahwa, dua buah matriks dapat dijumlahkan atau dikurangi apabila memiliki ukuran atau ordo yang sama.

Perhatikan bahwa matriks K L dan 2M memiliki ordo 2 cross times 2, dengan menjumlahakan elemen-elemen yang seletak maka didapatkan

K+L2M===(4102)+(1310)(6224)(4+161+320122+04)(1232)

Dengan demikian, nilai dari K plus L minus 2 M adalah (1232).

0

Roboguru

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