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Jika matriks  dan , tentukan determinan .

Pertanyaan

Jika matriks A equals open parentheses table row 2 5 row 1 3 end table close parentheses dan B equals open parentheses table row 5 4 row 1 1 end table close parentheses, tentukan determinan open parentheses A B close parentheses to the power of negative 1 end exponent.

Pembahasan Soal:

Tentukan matriks A B.

table attributes columnalign right center left columnspacing 0px end attributes row cell A B end cell equals cell open parentheses table row 2 5 row 1 3 end table close parentheses open parentheses table row 5 4 row 1 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 2 times 5 plus 5 times 1 end cell cell 2 times 4 plus 5 times 1 end cell row cell 1 times 5 plus 3 times 1 end cell cell 1 times 4 plus 3 times 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row 15 13 row 8 7 end table close parentheses end cell end table 

Tentukan invers matriks A B.

table attributes columnalign right center left columnspacing 0px end attributes row cell A B to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator det space A B end fraction open parentheses table row 7 cell negative 13 end cell row cell negative 8 end cell 15 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 15 times 7 minus 13 times 8 end fraction open parentheses table row 7 cell negative 13 end cell row cell negative 8 end cell 15 end table close parentheses end cell row blank equals cell 1 over 1 open parentheses table row 7 cell negative 13 end cell row cell negative 8 end cell 15 end table close parentheses end cell row blank equals cell open parentheses table row 7 cell negative 13 end cell row cell negative 8 end cell 15 end table close parentheses end cell end table 

Sehingga determinannya:

table attributes columnalign right center left columnspacing 0px end attributes row cell det space A B to the power of negative 1 end exponent end cell equals cell 7 times 15 minus open parentheses negative 13 close parentheses times open parentheses negative 8 close parentheses end cell row blank equals cell 105 minus 104 end cell row blank equals 1 end table 

Jadi, determinan open parentheses A B close parentheses to the power of negative 1 end exponent adalah 1.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

F. Nur

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Terakhir diupdate 05 Juni 2021

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Pertanyaan yang serupa

Determinan matriks  yang memenuhi persamaan  adalah ...

Pembahasan Soal:

Persamaan matriks bentuk A times X equals B yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell A times X end cell equals B row X equals cell A to the power of negative 1 end exponent B end cell end table

Jika terdapat matriks A equals open parentheses table row a b row c d end table close parentheses, inversnya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row a b row c d end table close parentheses end cell row cell A to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator det space A end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator a times d minus b times c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses end cell end table

Jika terdapat matiks A equals open parentheses table row a b row c d end table close parentheses, determinannya yaitu;

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row a b row c d end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator a times d minus b times c end fraction end cell end table

Perhitungan pada soal, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row 4 7 row 3 5 end table close parentheses times K end cell equals cell open parentheses table row 3 1 row 2 1 end table close parentheses end cell row K equals cell open parentheses table row 4 7 row 3 5 end table close parentheses to the power of negative 1 end exponent times open parentheses table row 3 1 row 2 1 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 4 open parentheses 5 close parentheses minus 7 open parentheses 3 close parentheses end fraction open parentheses table row 5 cell negative 7 end cell row cell negative 3 end cell 4 end table close parentheses times open parentheses table row 3 1 row 2 1 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 20 minus 21 end fraction open parentheses table row 5 cell negative 7 end cell row cell negative 3 end cell 4 end table close parentheses times open parentheses table row 3 1 row 2 1 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 1 end fraction open parentheses table row cell 5 open parentheses 3 close parentheses plus open parentheses negative 7 close parentheses open parentheses 2 close parentheses end cell cell 5 open parentheses 1 close parentheses plus open parentheses negative 7 close parentheses open parentheses 1 close parentheses end cell row cell negative 3 open parentheses 3 close parentheses plus 4 open parentheses 2 close parentheses end cell cell negative 3 open parentheses 1 close parentheses plus 4 open parentheses 1 close parentheses end cell end table close parentheses end cell row blank equals cell negative 1 open parentheses table row cell 15 minus 14 end cell cell 5 minus 7 end cell row cell negative 9 plus 8 end cell cell negative 3 plus 4 end cell end table close parentheses end cell row blank equals cell negative 1 open parentheses table row 1 cell negative 2 end cell row cell negative 1 end cell 1 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell 2 row 1 cell negative 1 end cell end table close parentheses end cell end table

Kemudian tentukan nilai determinan dari matriks K.

table attributes columnalign right center left columnspacing 0px end attributes row K equals cell open parentheses table row cell negative 1 end cell 2 row 1 cell negative 1 end cell end table close parentheses end cell row cell det space K end cell equals cell fraction numerator 1 over denominator negative 1 open parentheses negative 1 close parentheses minus 2 open parentheses 1 close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator 1 minus 2 end fraction end cell row blank equals cell fraction numerator 1 over denominator negative 1 end fraction end cell row blank equals cell negative 1 end cell end table

Determinan matriks K yang memenuhi persamaan open parentheses table row 4 7 row 3 5 end table close parentheses times K equals open parentheses table row 3 1 row 2 1 end table close parentheses adalah negative 1.

Jadi, jawaban yang tepat adalah C.

0

Roboguru

Misal A adalah matriks berordo dan menyatakan invers dari A, menyatakan determinan A, dan k adalah sebuah konstanta. Buktikan bahwa determinan dari .

Pembahasan Soal:

Ingat rumus determinan dan invers matriks berikut:

1. A equals open parentheses table row a b row c d end table close parentheses rightwards arrow open vertical bar A close vertical bar equals a times d minus b times c

2. A equals open parentheses table row a b row c d end table close parentheses rightwards arrow A to the power of negative 1 end exponent equals fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses

Akan dibuktikan bahwa open vertical bar open parentheses k A close parentheses to the power of negative 1 end exponent close vertical bar equals fraction numerator 1 over denominator k squared open vertical bar A close vertical bar end fraction

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell k A end cell equals cell k open parentheses table row a b row c d end table close parentheses end cell row blank equals cell open parentheses table row cell k a end cell cell k b end cell row cell k c end cell cell k d end cell end table close parentheses end cell row cell open parentheses k A close parentheses to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator k a times k d minus k b times k c end fraction open parentheses table row cell k d end cell cell negative k b end cell row cell negative k c end cell cell k a end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator k squared left parenthesis a d minus b c right parenthesis end fraction open parentheses table row cell k d end cell cell negative k b end cell row cell negative k c end cell cell k a end cell end table close parentheses end cell row blank equals cell open parentheses table row cell fraction numerator k d over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell cell fraction numerator negative k b over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell row cell fraction numerator negative k c over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell cell fraction numerator k a over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell end table close parentheses end cell row blank equals cell open parentheses table row cell fraction numerator d over denominator k left parenthesis a d minus b c right parenthesis end fraction end cell cell fraction numerator negative b over denominator k left parenthesis a d minus b c right parenthesis end fraction end cell row cell fraction numerator negative c over denominator k left parenthesis a d minus b c right parenthesis end fraction end cell cell fraction numerator a over denominator k left parenthesis a d minus b c right parenthesis end fraction end cell end table close parentheses end cell row cell open vertical bar open parentheses k A close parentheses to the power of negative 1 end exponent close vertical bar end cell equals cell fraction numerator d over denominator k left parenthesis a d minus b c right parenthesis end fraction times fraction numerator a over denominator k left parenthesis a d minus b c right parenthesis end fraction minus open parentheses fraction numerator negative b over denominator k left parenthesis a d minus b c right parenthesis end fraction close parentheses open parentheses fraction numerator negative c over denominator k left parenthesis a d minus b c right parenthesis end fraction close parentheses end cell row blank equals cell fraction numerator a d over denominator k squared left parenthesis a d minus b c right parenthesis squared end fraction minus fraction numerator b c over denominator k squared left parenthesis a d minus b c right parenthesis squared end fraction end cell row blank equals cell fraction numerator a d minus b c over denominator k squared left parenthesis a d minus b c right parenthesis squared end fraction end cell row blank equals cell fraction numerator 1 over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell row blank equals cell fraction numerator 1 over denominator k squared open vertical bar A close vertical bar end fraction end cell end table end style

Jadi, terbukti bahwa open vertical bar open parentheses k A close parentheses to the power of negative 1 end exponent close vertical bar equals fraction numerator 1 over denominator k squared open vertical bar A close vertical bar end fraction.

0

Roboguru

Diketahui matriks . Jika  invers matriks  dan  adalah invers matriks , maka deteminan matriks  adalah ...

Pembahasan Soal:

Diketahui:

A equals open parentheses table row 1 cell negative 1 end cell row 5 cell negative 4 end cell end table close parentheses comma thin space B equals open parentheses table row 7 3 row 2 1 end table close parentheses

Mencari invers matriks A:

table attributes columnalign right center left columnspacing 0px end attributes row cell A to the power of negative 1 end exponent end cell equals cell open parentheses table row 1 cell negative 1 end cell row 5 cell negative 4 end cell end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell fraction numerator 1 over denominator det open parentheses table row 1 cell negative 1 end cell row 5 cell negative 4 end cell end table close parentheses end fraction open parentheses table row cell negative 4 end cell cell negative open parentheses negative 1 close parentheses end cell row cell negative 5 end cell 1 end table close parentheses end cell row blank equals cell 1 over 1 open parentheses table row cell negative 4 end cell cell negative open parentheses negative 1 close parentheses end cell row cell negative 5 end cell 1 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 4 end cell 1 row cell negative 5 end cell 1 end table close parentheses end cell end table 

Mencari invers matriks B:

table attributes columnalign right center left columnspacing 0px end attributes row cell B to the power of negative 1 end exponent end cell equals cell open parentheses table row 7 3 row 2 1 end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell fraction numerator 1 over denominator det open parentheses table row 7 3 row 2 1 end table close parentheses end fraction open parentheses table row 1 cell negative 3 end cell row cell negative 2 end cell 7 end table close parentheses end cell row blank equals cell 1 over 1 open parentheses table row 1 cell negative 3 end cell row cell negative 2 end cell 7 end table close parentheses end cell row blank equals cell open parentheses table row 1 cell negative 3 end cell row cell negative 2 end cell 7 end table close parentheses end cell end table 

Mencari determinan matriks B to the power of negative 1 end exponent A to the power of negative 1 end exponent:

det open parentheses open parentheses table row 1 cell negative 3 end cell row cell negative 2 end cell 7 end table close parentheses open parentheses table row cell negative 4 end cell 1 row cell negative 5 end cell 1 end table close parentheses close parentheses equals det open parentheses table row 11 cell negative 2 end cell row cell negative 27 end cell 5 end table close parentheses equals 11 times 5 minus open parentheses negative 2 close parentheses open parentheses negative 27 close parentheses equals 1  

Jadi, determinan matriks B to the power of negative 1 end exponent A to the power of negative 1 end exponent adalah 1.

Dengan demikian, jawaban yang tepat adalah D.

0

Roboguru

Diketahui matriks , maka determinan matriks  adalah...

Pembahasan Soal:

Ingatlah definisi dan sifat-sifat determinan matriks berikut:

rightwards double arrow P equals open parentheses table row a b row c d end table close parentheses rightwards arrow open vertical bar P close vertical bar equals a d minus b c rightwards double arrow open vertical bar P Q close vertical bar equals open vertical bar P close vertical bar open vertical bar Q close vertical bar rightwards double arrow open vertical bar P to the power of negative 1 end exponent close vertical bar equals fraction numerator 1 over denominator open vertical bar P close vertical bar end fraction

Berdasarkan sifat di atas, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar P to the power of negative 1 end exponent Q to the power of negative 1 end exponent close vertical bar end cell equals cell open vertical bar P to the power of negative 1 end exponent close vertical bar open vertical bar Q to the power of negative 1 end exponent close vertical bar end cell row blank equals cell fraction numerator 1 over denominator open vertical bar P close vertical bar open vertical bar Q close vertical bar end fraction end cell row blank equals cell fraction numerator 1 over denominator open parentheses 2 times 3 minus 5 times 1 close parentheses open parentheses 5 times 1 minus 4 times 1 close parentheses end fraction end cell row blank equals cell 1 over 1 end cell row blank equals 1 end table

Jadi, jawaban yang tepat adalah C.

0

Roboguru

Jika  merupakan matriks yang mempunyai invers, maka hasil kali semua nilai  yang mungkin sehingga  adalah ....

Pembahasan Soal:

Ingat kembali konsep sifat determinan pada matriks sebagai berikut:

det space open parentheses straight A to the power of negative 1 end exponent close parentheses equals fraction numerator 1 over denominator det space open parentheses straight A close parentheses end fraction

Oleh karena itu, Jika A equals open parentheses table row a 3 row 1 1 end table close parentheses merupakan matriks yang mempunyai invers, maka hasil kali semua nilai a yang mungkin sehingga 2 space det open parentheses straight A close parentheses equals det open parentheses straight A to the power of negative 1 end exponent close parentheses plus 1 adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 space det open parentheses straight A close parentheses end cell equals cell det open parentheses straight A to the power of negative 1 end exponent close parentheses plus 1 end cell row cell 2 open parentheses a.1 minus 3.1 close parentheses end cell equals cell fraction numerator 1 over denominator a.1 minus 3.1 end fraction plus 1 end cell row cell 2 open parentheses a minus 3 close parentheses end cell equals cell fraction numerator 1 over denominator a minus 3 end fraction plus 1 end cell row cell 2 open parentheses a minus 3 close parentheses open parentheses a minus 3 close parentheses end cell equals cell 1 plus open parentheses a minus 3 close parentheses end cell row cell 2 open parentheses a squared minus 6 a plus 9 close parentheses end cell equals cell 1 plus a minus 3 end cell row cell 2 a squared minus 12 a plus 18 end cell equals cell a minus 2 end cell row cell 2 a squared minus 12 a plus 18 minus a plus 2 end cell equals 0 row cell 2 a squared minus 13 a plus 20 end cell equals 0 row cell open parentheses 2 a minus 5 close parentheses open parentheses a minus 4 close parentheses end cell equals 0 end table

a equals 5 over 2 comma space a equals 4

Hasil kali nilai a adalah 5 over 2 cross times 4 equals 10.

Dengan demikian, hasil kali nilai a yang mungkin adalah 10.

Jadi, tidak ada pilihan jawaban yang tepat.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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