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Jika P= ⎝ ⎛ ​ p 0 0 ​ 1 p 0 ​ 0 1 p ​ ⎠ ⎞ ​ , Tenukan matriks P n !

Jika , Tenukan matriks !

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S. Solehuzain

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

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Pembahasan

Diketahui Ingat Perkalian matriks Untuk mencari rumus matriks , terlebih dahulu mencari pola untuk . Sehingga diperoleh perhitungan Matriks Matriks Matriks Dengan demikian, dari beberapa matrik diatas dipeoleh pola untuk matriks

Diketahui text P= end text open parentheses table row p 1 0 row 0 p 1 row 0 0 p end table close parentheses

Ingat Perkalian matriks

A times B equals open parentheses table row a b row c d end table close parentheses times open parentheses table row e f row g h end table close parentheses equals open parentheses table row cell a e plus b g end cell cell a f plus b h end cell row cell c e plus d g end cell cell c f plus d h end cell end table close parentheses 

Untuk mencari rumus matriks P to the power of n, terlebih dahulu mencari pola untuk P squared comma P cubed space text dan end text space P to the power of 4.

Sehingga diperoleh perhitungan 

  • Matriks P squared

table attributes columnalign right center left columnspacing 0px end attributes row cell P squared end cell equals cell P. P end cell row blank equals cell open parentheses table row p 1 0 row 0 p 1 row 0 0 p end table close parentheses times open parentheses table row p 1 0 row 0 p 1 row 0 0 p end table close parentheses end cell row blank equals cell open parentheses table row cell p. p plus 1.0 plus 0.0 end cell cell p.1 plus 1. p plus 0.0 end cell cell p.0 plus 1.1 plus 0. p end cell row cell 0. p plus p.0 plus 1.0 end cell cell 0.1 plus p. p plus 1.0 end cell cell 0.0 plus p.1 plus 1. p end cell row cell 0. p plus 0.0 plus p.0 end cell cell 0.1 plus 0. p plus p.0 end cell cell 0.0 plus 0.1 plus p. p end cell end table close parentheses end cell row blank equals cell open parentheses table row cell p squared end cell cell 2 p end cell 1 row 0 cell p squared end cell cell 2 p end cell row 0 0 cell p squared end cell end table close parentheses end cell end table

  • Matriks P cubed

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell P cubed end cell equals cell P squared. P end cell row blank equals cell open parentheses table row cell p squared end cell cell 2 p end cell 1 row 0 cell p squared end cell cell 2 p end cell row 0 0 cell p squared end cell end table close parentheses times open parentheses table row p 1 0 row 0 p 1 row 0 0 p end table close parentheses end cell row blank equals cell open parentheses table row cell p squared. p plus 2 p.0 plus 1.0 end cell cell p squared.1 plus 2 p. p plus 1.0 end cell cell p squared.0 plus 2 p.1 plus 1. p end cell row cell 0. p plus p squared.0 plus 2 p.0 end cell cell 0.1 plus p squared. p plus 2 p.0 end cell cell 0.0 plus p squared.1 plus 2 p. p end cell row cell 0. p plus 0.0 plus p squared.0 end cell cell 0.1 plus 0. p plus p squared.0 end cell cell 0.0 plus 0.1 plus p squared. p end cell end table close parentheses end cell row blank equals cell open parentheses table row cell p cubed end cell cell 3 p squared end cell cell 3 p end cell row 0 cell p cubed end cell cell 3 p squared end cell row 0 0 cell p cubed end cell end table close parentheses end cell end table end style

  • Matriks P to the power of 4

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell P to the power of 4 end cell equals cell P cubed. P end cell row blank equals cell open parentheses table row cell p cubed end cell cell 3 p squared end cell cell 3 p end cell row 0 cell p cubed end cell cell 3 p squared end cell row 0 0 cell p cubed end cell end table close parentheses times open parentheses table row p 1 0 row 0 p 1 row 0 0 p end table close parentheses end cell row blank equals cell open parentheses table row cell p cubed. p plus 3 p squared.0 plus 3 p.0 end cell cell p cubed.1 plus 3 p squared. p plus 3 p.0 end cell cell p cubed.0 plus 3 p squared.1 plus 3 p. p end cell row cell 0. p plus p cubed.0 plus 3 p squared.0 end cell cell 0.1 plus p cubed. p plus 3 p squared.0 end cell cell 0.0 plus p cubed.1 plus 3 p squared. p end cell row cell 0. p plus 0.0 plus p cubed.0 end cell cell 0.1 plus 0. p plus p cubed.0 end cell cell 0.0 plus 0.1 plus p cubed. p end cell end table close parentheses end cell row blank equals cell open parentheses table row cell p to the power of 4 end cell cell 4 p cubed end cell cell 3 p squared end cell row 0 cell p to the power of 4 end cell cell 4 p cubed end cell row 0 0 cell p to the power of 4 end cell end table close parentheses end cell end table end style

Dengan demikian, dari beberapa matrik diatas dipeoleh pola untuk matriks P to the power of n equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell p to the power of n end cell cell n p to the power of n minus 1 end exponent end cell cell n minus 1 p to the power of n minus 2 end exponent end cell row 0 cell p to the power of n end cell cell n p to the power of n minus 1 end exponent end cell row 0 0 cell p to the power of n end cell end table close parentheses end cell end table    

 

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