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Diketahui matriks A=(U1​U4​​−U2​U3​​) dan Un​ adalah suku ke-n barisan geometri. Jika U1​+U3​=p1​  dan U2​+U4​=q1​ dengan p,q=0 maka determinan A sama dengan...

Pertanyaan

Diketahui matriks A equals open parentheses table row cell U subscript 1 end cell cell negative U subscript 2 end cell row cell U subscript 4 end cell cell U subscript 3 end cell end table close parentheses dan straight U subscript straight n adalah suku ke-barisan geometri. Jika U subscript 1 plus U subscript 3 equals 1 over straight p  dan U subscript 2 plus U subscript 4 equals 1 over straight q dengan straight p comma straight q not equal to 0 maka determinan A sama dengan...

  1. fraction numerator 1 over denominator straight p squared plus straight q squared end fraction

  2. fraction numerator straight p squared over denominator straight p squared plus straight q squared end fraction

  3. fraction numerator pq squared over denominator straight p squared plus straight q squared end fraction

  4. fraction numerator 1 over denominator straight p plus straight q end fraction

  5. fraction numerator straight q over denominator straight p plus straight q end fraction

O. Rahmawati

Master Teacher

Mahasiswa/Alumni UIN Sunan Gunung Djati Bandung

Jawaban terverifikasi

Pembahasan

Diketahui: 

  • A equals open parentheses table row cell U subscript 1 end cell cell negative U subscript 2 end cell row cell U subscript 4 end cell cell U subscript 3 end cell end table close parentheses
  • U subscript 1 plus U subscript 3 equals 1 over straight p
  • U subscript 2 plus U subscript 4 equals 1 over straight q

Ditanya: det left parenthesis A right parenthesis equals...

Jawab:

  • Menentukan rasio:

​​​​​​​table attributes columnalign right center left columnspacing 0px end attributes row cell U subscript 1 plus U subscript 3 end cell equals cell 1 over straight p end cell row cell a plus a r squared end cell equals cell 1 over straight p end cell row cell a open parentheses 1 plus r squared close parentheses end cell equals cell 1 over straight p comma space straight p not equal to 0 end cell end table​​​​​​​

dan 

U subscript 2 plus U subscript 4 equals 1 over straight q a r plus a r cubed equals 1 over straight q a r open parentheses 1 plus r squared close parentheses equals 1 over straight q comma space straight q not equal to 0

Jadi, rasionya adalah

fraction numerator a r open parentheses 1 plus r squared close parentheses over denominator a open parentheses 1 plus r squared close parentheses end fraction equals fraction numerator begin display style 1 over straight q end style over denominator begin display style 1 over straight p end style end fraction equals straight p over straight q

  • Menentukan nilai suku pertama atau a:

table attributes columnalign right center left columnspacing 0px end attributes row cell a open parentheses 1 plus r squared close parentheses end cell equals cell 1 over straight p end cell row a equals cell fraction numerator 1 over denominator straight p open parentheses 1 plus r squared close parentheses end fraction end cell end table

  • Menentukan determinan A:

​​​​​​​table attributes columnalign right center left columnspacing 0px end attributes row cell d e t left parenthesis A right parenthesis end cell equals cell U subscript 1 times U subscript 3 minus open parentheses negative U subscript 2 close parentheses times U subscript 4 end cell row blank equals cell U subscript 1 times U subscript 3 plus U subscript 2 times U subscript 4 end cell row blank equals cell a times a r squared plus a r times a r cubed end cell row blank equals cell a squared r squared open parentheses 1 plus r squared close parentheses end cell row blank equals cell open parentheses fraction numerator 1 over denominator straight p open parentheses 1 plus r squared close parentheses end fraction close parentheses squared r squared open parentheses 1 plus r squared close parentheses end cell row blank equals cell fraction numerator r squared over denominator straight p squared open parentheses 1 plus r squared close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses begin display style straight p over straight q end style close parentheses squared over denominator straight p squared open parentheses 1 plus open parentheses begin display style straight p over straight q end style close parentheses squared close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator straight q squared open parentheses 1 plus begin display style straight p squared over straight q squared end style close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator straight p squared plus straight q squared end fraction end cell row blank blank blank end table

Oleh karena itu jawbaan yang benar adalah A.

 

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