Diberikan matriks A dan B berordo 2×2 dengan A+B=(0−41−3) danAB+BA=(−121200). Matriks A2+B2=....

Pertanyaan

Diberikan matriks A dan B berordo $\style{font-size:14px}{2\times2}$ dengan $\style{font-size:14px}{A+B=\begin{pmatrix}0&1\\-4&-3\end{pmatrix}\text{ dan}\;AB+BA=\begin{pmatrix}-12&0\\12&0\end{pmatrix}}$ . Matriks $A^2+B^2=....$

$begin mathsize 14px style open parentheses table row 8 cell negative 3 end cell row 0 5 end table close parentheses end style$
$begin mathsize 14px style open parentheses table row 5 3 row 0 8 end table close parentheses end style$
$begin mathsize 14px style open parentheses table row cell negative 12 end cell cell negative 1 end cell row 16 3 end table close parentheses end style$
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Pembahasan:

Ingat kembali: Jika matriks $A equals open parentheses table row a b row c d end table close parentheses$ dan $B equals open parentheses table row m n row o p end table close parentheses$ , maka

$A B equals open parentheses table row a b row c d end table close parentheses open parentheses table row m n row o p end table close parentheses equals open parentheses table row cell a m plus b o end cell cell a n plus b p end cell row cell c m plus d o end cell cell c n plus d p end cell end table close parentheses$
$table attributes columnalign right center left columnspacing 0px end attributes row cell A squared plus A B plus B A plus B squared end cell equals cell open parentheses A plus B close parentheses open parentheses A plus B close parentheses end cell end table$
$A plus B equals B plus A$

Dengan demikian,

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell A squared plus A B plus B A plus B squared end cell equals cell open parentheses A plus B close parentheses open parentheses A plus B close parentheses end cell row cell A squared plus B squared plus A B plus B A end cell equals cell open parentheses A plus B close parentheses open parentheses A plus B close parentheses end cell row cell A squared plus B squared plus open parentheses table row cell negative 12 end cell 0 row 12 0 end table close parentheses end cell equals cell open parentheses table row 0 1 row cell negative 4 end cell cell negative 3 end cell end table close parentheses open parentheses table row 0 1 row cell negative 4 end cell cell negative 3 end cell end table close parentheses end cell row cell A squared plus B squared plus open parentheses table row cell negative 12 end cell 0 row 12 0 end table close parentheses end cell equals cell open parentheses table row cell 0 open parentheses 0 close parentheses plus 1 open parentheses negative 4 close parentheses end cell cell 0 open parentheses 1 close parentheses plus 1 open parentheses negative 3 close parentheses end cell row cell open parentheses negative 4 close parentheses open parentheses 0 close parentheses plus open parentheses negative 3 close parentheses open parentheses negative 4 close parentheses end cell cell open parentheses negative 4 close parentheses open parentheses 1 close parentheses plus open parentheses negative 3 close parentheses open parentheses negative 3 close parentheses end cell end table close parentheses end cell row cell A squared plus B squared plus open parentheses table row cell negative 12 end cell 0 row 12 0 end table close parentheses end cell equals cell open parentheses table row cell 0 plus open parentheses negative 4 close parentheses end cell cell 0 plus open parentheses negative 3 close parentheses end cell row cell 0 plus 12 end cell cell open parentheses negative 4 close parentheses plus 9 end cell end table close parentheses end cell row cell A squared plus B squared plus open parentheses table row cell negative 12 end cell 0 row 12 0 end table close parentheses end cell equals cell open parentheses table row cell negative 4 end cell cell negative 3 end cell row 12 5 end table close parentheses end cell row cell A squared plus B squared end cell equals cell open parentheses table row cell negative 4 end cell cell negative 3 end cell row 12 5 end table close parentheses minus open parentheses table row cell negative 12 end cell 0 row 12 0 end table close parentheses end cell row cell A squared plus B squared end cell equals cell open parentheses table row cell negative 4 minus open parentheses negative 12 close parentheses end cell cell negative 3 minus 0 end cell row cell 12 minus 12 end cell cell 5 minus 0 end cell end table close parentheses end cell row cell A squared plus B squared end cell equals cell open parentheses table row 8 cell negative 3 end cell row 0 5 end table close parentheses end cell end table end style$

Jadi, matriks $A squared plus B squared equals open parentheses table row 8 cell negative 3 end cell row 0 5 end table close parentheses$ .

Oleh karena itu, jawaban yang benar adalah A. $space$

Jawaban terverifikasi

Dijawab oleh:

I. Ridha

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 06 Oktober 2021