Pertanyaan

Jika matriks ( a 4 − 3 b ) mentransformasikan titik ( 5 , 1 ) ke titik ( 7 , 12 ) dan inversnya mentransformasikan titik P ke titik ( 1 , 0 ) , maka koordinat titik adalah ....

Jika matriks $(a 4 - 3 b)$ mentransformasikan titik $(5, 1)$ ke titik $(7, 12)$ dan inversnya mentransformasikan titik $P$ ke titik $(1, 0)$ , maka koordinat titik straight P adalah ....

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah B.

Pembahasan

Ingat kembali: Diketahuimatriks mentransformasikan titik ke titik . Maka: didapat: dan matriksnya: inversnya: Kemudianinvers matriks di atasditransformasikan titik ke titik , sehingga: Jadi, jawaban yang benar adalah B.

Ingat kembali:

$table attributes columnalign right center left columnspacing 0px end attributes row straight A equals cell open parentheses table row straight a straight b row straight c straight d end table close parentheses rightwards arrow straight A to the power of negative 1 end exponent equals fraction numerator 1 over denominator ad minus bc end fraction open parentheses table row straight d cell negative straight b end cell row cell negative straight c end cell straight a end table close parentheses end cell row AX equals cell straight B rightwards arrow straight X equals straight A to the power of negative 1 end exponent times straight B end cell end table$

Diketahui matriks open parentheses table row a cell negative 3 end cell row 4 b end table close parentheses mentransformasikan titik open parentheses 5 comma space 1 close parentheses ke titik open parentheses 7 comma space 12 close parentheses . Maka:

didapat:

table attributes columnalign right center left columnspacing 0px end attributes row cell 5 a minus 3 end cell equals 7 row cell 5 a end cell equals cell 7 plus 3 end cell row cell 5 a end cell equals 10 row a equals cell 10 over 5 end cell row a equals 2 end table dan

matriksnya:

open parentheses table row a cell negative 3 end cell row 4 b end table close parentheses rightwards arrow open parentheses table row 2 cell negative 3 end cell row 4 cell negative 8 end cell end table close parentheses

inversnya:

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row 2 cell negative 3 end cell row 4 cell negative 8 end cell end table close parentheses to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator 2 times open parentheses negative 8 close parentheses minus open parentheses negative 3 close parentheses times 4 end fraction open parentheses table row cell negative 8 end cell 3 row cell negative 4 end cell 2 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 16 plus 12 end fraction open parentheses table row cell negative 8 end cell 3 row cell negative 4 end cell 2 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 4 end fraction open parentheses table row cell negative 8 end cell 3 row cell negative 4 end cell 2 end table close parentheses end cell row blank equals cell open parentheses table row 2 cell negative 3 over 4 end cell row 1 cell negative 1 half end cell end table close parentheses end cell end table$

Kemudian invers matriks di atas ditransformasikan titik straight P ke titik open parentheses 1 comma space 0 close parentheses , sehingga:

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row 2 cell negative 3 over 4 end cell row 1 cell negative 1 half end cell end table close parentheses open parentheses table row x row y end table close parentheses end cell row cell open parentheses table row 1 row 0 end table close parentheses end cell equals cell open parentheses table row 2 cell negative 3 over 4 end cell row 1 cell negative 1 half end cell end table close parentheses open parentheses table row x row y end table close parentheses end cell row cell open parentheses table row 2 cell negative 3 over 4 end cell row 1 cell negative 1 half end cell end table close parentheses open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row 1 row 0 end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row 2 cell negative 3 over 4 end cell row 1 cell negative 1 half end cell end table close parentheses to the power of negative 1 end exponent open parentheses table row 1 row 0 end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell fraction numerator 1 over denominator 2 open parentheses negative begin display style 1 half end style close parentheses minus open parentheses negative begin display style 3 over 4 end style close parentheses 1 end fraction open parentheses table row cell negative 1 half end cell cell 3 over 4 end cell row cell negative 1 end cell 2 end table close parentheses open parentheses table row 1 row 0 end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell fraction numerator 1 over denominator negative 1 plus begin display style 3 over 4 end style end fraction open parentheses table row cell negative 1 half end cell cell 3 over 4 end cell row cell negative 1 end cell 2 end table close parentheses open parentheses table row 1 row 0 end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell fraction numerator 1 over denominator negative begin display style 1 fourth end style end fraction open parentheses table row cell negative 1 half end cell cell 3 over 4 end cell row cell negative 1 end cell 2 end table close parentheses open parentheses table row 1 row 0 end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell negative 4 open parentheses table row cell negative 1 half end cell cell 3 over 4 end cell row cell negative 1 end cell 2 end table close parentheses open parentheses table row 1 row 0 end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row 2 cell negative 3 end cell row 4 cell negative 8 end cell end table close parentheses open parentheses table row 1 row 0 end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row cell 2 minus 0 end cell row cell 4 minus 0 end cell end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row 2 row 4 end table close parentheses end cell end table$

Jadi, jawaban yang benar adalah B.