Pertanyaan

P adalah matriks berordo 2 × 2 yang memenuhi persamaan: P ( 2 − 1 4 3 ) = ( 5 8 15 − 4 ) Tentukan ∣ P ∣ .

$P$ adalah matriks berordo $2 \times 2$ yang memenuhi persamaan:

$P (2 - 1 43) = (58 15 - 4)$

Tentukan $∣ P ∣$ .

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S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

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Pembahasan

Persamaan matriks bentuk berlaku: Invers matriks berlaku: Determinan matriks berlaku: Diperoleh penyelesaiannya sebagai berikut: Kemudian tentukan determinan dari matriks . Dengan demikian, adalah .

Persamaan matriks bentuk X A equals B berlaku:

Invers matriks 2 cross times 2 berlaku:

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

Determinan matriks 2 cross times 2 berlaku:

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 open vertical bar A close vertical bar end cell equals cell open vertical bar table row a b row c d end table close vertical bar end cell row blank equals cell a d minus b c end cell end table

Diperoleh penyelesaiannya sebagai berikut:

$table attributes columnalign right center left columnspacing 0px end attributes row cell P open parentheses table row 2 4 row cell negative 1 end cell 3 end table close parentheses end cell equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses end cell row P equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses open parentheses table row 2 4 row cell negative 1 end cell 3 end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses times fraction numerator 1 over denominator 2 open parentheses 3 close parentheses minus 4 open parentheses negative 1 close parentheses end fraction open parentheses table row 3 cell negative 4 end cell row 1 2 end table close parentheses end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses times fraction numerator 1 over denominator 6 plus 4 end fraction open parentheses table row 3 cell negative 4 end cell row 1 2 end table close parentheses end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses times 1 over 10 open parentheses table row 3 cell negative 4 end cell row 1 2 end table close parentheses end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses open parentheses table row cell 1 over 10 times 3 end cell cell 1 over 10 times open parentheses negative 4 close parentheses end cell row cell 1 over 10 times 1 end cell cell 1 over 10 times 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses open parentheses table row cell 3 over 10 end cell cell negative 4 over 10 end cell row cell 1 over 10 end cell cell 2 over 10 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 5 open parentheses 3 over 10 close parentheses plus 15 open parentheses 1 over 10 close parentheses end cell cell 5 open parentheses negative 4 over 10 close parentheses plus 15 open parentheses 2 over 10 close parentheses end cell row cell 8 open parentheses 3 over 10 close parentheses plus open parentheses negative 4 close parentheses open parentheses 1 over 10 close parentheses end cell cell 8 open parentheses negative 4 over 10 close parentheses plus open parentheses negative 4 close parentheses open parentheses 2 over 10 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 15 over 10 plus 15 over 10 end cell cell negative 20 over 10 plus 30 over 10 end cell row cell 24 over 10 minus 4 over 10 end cell cell negative 32 over 10 minus 8 over 10 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 30 over 10 end cell cell 10 over 10 end cell row cell 20 over 10 end cell cell negative 40 over 10 end cell end table close parentheses end cell row blank equals cell open parentheses table row 3 1 row 2 cell negative 4 end cell end table close parentheses end cell end table$

Kemudian tentukan determinan dari matriks .

Dengan demikian, open vertical bar P close vertical bar adalah negative 14 .