Berikut ini adalah pertanyaan dari AghaJaiza pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas
Diketahui matriks A = ![\left[\begin{array}{ccc}4&&-2\\&&\\3&&1\end{array}\right]](https://tex.z-dn.net/?f=%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D4%26%26-2%5C%5C%26%26%5C%5C3%26%261%5Cend%7Barray%7D%5Cright%5D%20 "\left[\begin{array}{ccc}4&&-2\\&&\\3&&1\end{array}\right]")
dan B = ![\left[\begin{array}{ccc}2&&4\\&&\\-5&&-1\end{array}\right]](https://tex.z-dn.net/?f=%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26%264%5C%5C%26%26%5C%5C-5%26%26-1%5Cend%7Barray%7D%5Cright%5D%20 "\left[\begin{array}{ccc}2&&4\\&&\\-5&&-1\end{array}\right]")
a. Tentukan hasil A² - B² !
Jawaban dan Penjelasan
Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.
Dengan pemangkatan matriks mengartikan:
![$\begin{align}&A^2-B^2 \\ &=\left[\begin{array}{cc}4&-2\\3&1\end{array}\right]^2-\left[\begin{array}{cc}2&4\\-5&-1\end{array}\right]^2 \\ &=\left[\begin{array}{cc}4&-2\\3&1\end{array}\right]\left[\begin{array}{cc}4&-2\\3&1\end{array}\right]-\left[\begin{array}{cc}2&4\\-5&-1\end{array}\right]\left[\begin{array}{cc}2&4\\-5&-1\end{array}\right] \\ \end{align}](https://tex.z-dn.net/?f=%24%5Cbegin%7Balign%7D%26A%5E2-B%5E2%20%5C%5C%20%26%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4%26-2%5C%5C3%261%5Cend%7Barray%7D%5Cright%5D%5E2-%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2%264%5C%5C-5%26-1%5Cend%7Barray%7D%5Cright%5D%5E2%20%5C%5C%20%26%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4%26-2%5C%5C3%261%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4%26-2%5C%5C3%261%5Cend%7Barray%7D%5Cright%5D-%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2%264%5C%5C-5%26-1%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2%264%5C%5C-5%26-1%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%20%5Cend%7Balign%7D "$\begin{align}&A^2-B^2 \\ &=\left[\begin{array}{cc}4&-2\\3&1\end{array}\right]^2-\left[\begin{array}{cc}2&4\\-5&-1\end{array}\right]^2 \\ &=\left[\begin{array}{cc}4&-2\\3&1\end{array}\right]\left[\begin{array}{cc}4&-2\\3&1\end{array}\right]-\left[\begin{array}{cc}2&4\\-5&-1\end{array}\right]\left[\begin{array}{cc}2&4\\-5&-1\end{array}\right] \\ \end{align}")
![$\begin{align}&=\left[\begin{array}{cc}4.4+(-2).3&4.(-2)+(-2).1\\3.4+1.3&3.(-2)+1.1\end{array}\right]-\ddots \\ &\left[\begin{array}{cc}2.2+4.(-5)&2.4+4.(-1)\\(-5).2+(-1).(-5)&(-5).4+(-1).(-1)\end{array}\right] \\ &=\left[\begin{array}{cc}16-6&-8-2\\12+3&-6+1\end{array}\right]-\left[\begin{array}{cc}4-20&8-4\\-10+5&-20+1\end{array}\right] \\ \end{align}](https://tex.z-dn.net/?f=%24%5Cbegin%7Balign%7D%26%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4.4%2B%28-2%29.3%264.%28-2%29%2B%28-2%29.1%5C%5C3.4%2B1.3%263.%28-2%29%2B1.1%5Cend%7Barray%7D%5Cright%5D-%5Cddots%20%5C%5C%20%26%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2.2%2B4.%28-5%29%262.4%2B4.%28-1%29%5C%5C%28-5%29.2%2B%28-1%29.%28-5%29%26%28-5%29.4%2B%28-1%29.%28-1%29%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%20%26%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D16-6%26-8-2%5C%5C12%2B3%26-6%2B1%5Cend%7Barray%7D%5Cright%5D-%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4-20%268-4%5C%5C-10%2B5%26-20%2B1%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%20%5Cend%7Balign%7D "$\begin{align}&=\left[\begin{array}{cc}4.4+(-2).3&4.(-2)+(-2).1\\3.4+1.3&3.(-2)+1.1\end{array}\right]-\ddots \\ &\left[\begin{array}{cc}2.2+4.(-5)&2.4+4.(-1)\\(-5).2+(-1).(-5)&(-5).4+(-1).(-1)\end{array}\right] \\ &=\left[\begin{array}{cc}16-6&-8-2\\12+3&-6+1\end{array}\right]-\left[\begin{array}{cc}4-20&8-4\\-10+5&-20+1\end{array}\right] \\ \end{align}")
![$\begin{align}&=\left[\begin{array}{cc}10&-10\\15&-5\end{array}\right]-\left[\begin{array}{cc}-16&4\\-5&-19\end{array}\right] \\ &=\left[\begin{array}{cc}10-(-16)&-10-4\\-15-(-5)&-5-(-19)\end{array}\right] \\ &=\left[\begin{array}{cc}6&-14\\-10&14\end{array}\right]\end{align}](https://tex.z-dn.net/?f=%24%5Cbegin%7Balign%7D%26%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D10%26-10%5C%5C15%26-5%5Cend%7Barray%7D%5Cright%5D-%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D-16%264%5C%5C-5%26-19%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%20%26%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D10-%28-16%29%26-10-4%5C%5C-15-%28-5%29%26-5-%28-19%29%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%20%26%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D6%26-14%5C%5C-10%2614%5Cend%7Barray%7D%5Cright%5D%5Cend%7Balign%7D "$\begin{align}&=\left[\begin{array}{cc}10&-10\\15&-5\end{array}\right]-\left[\begin{array}{cc}-16&4\\-5&-19\end{array}\right] \\ &=\left[\begin{array}{cc}10-(-16)&-10-4\\-15-(-5)&-5-(-19)\end{array}\right] \\ &=\left[\begin{array}{cc}6&-14\\-10&14\end{array}\right]\end{align}")
![Dengan pemangkatan matriks mengartikan:[tex]$\begin{align}&A^2-B^2 \\ &=\left[\begin{array}{cc}4&-2\\3&1\end{array}\right]^2-\left[\begin{array}{cc}2&4\\-5&-1\end{array}\right]^2 \\ &=\left[\begin{array}{cc}4&-2\\3&1\end{array}\right]\left[\begin{array}{cc}4&-2\\3&1\end{array}\right]-\left[\begin{array}{cc}2&4\\-5&-1\end{array}\right]\left[\begin{array}{cc}2&4\\-5&-1\end{array}\right] \\ \end{align}[/tex][tex]$\begin{align}&=\left[\begin{array}{cc}4.4+(-2).3&4.(-2)+(-2).1\\3.4+1.3&3.(-2)+1.1\end{array}\right]-\ddots \\ &\left[\begin{array}{cc}2.2+4.(-5)&2.4+4.(-1)\\(-5).2+(-1).(-5)&(-5).4+(-1).(-1)\end{array}\right] \\ &=\left[\begin{array}{cc}16-6&-8-2\\12+3&-6+1\end{array}\right]-\left[\begin{array}{cc}4-20&8-4\\-10+5&-20+1\end{array}\right] \\ \end{align}[/tex][tex]$\begin{align}&=\left[\begin{array}{cc}10&-10\\15&-5\end{array}\right]-\left[\begin{array}{cc}-16&4\\-5&-19\end{array}\right] \\ &=\left[\begin{array}{cc}10-(-16)&-10-4\\-15-(-5)&-5-(-19)\end{array}\right] \\ &=\left[\begin{array}{cc}6&-14\\-10&14\end{array}\right]\end{align}[/tex]](https://id-static.z-dn.net/files/db0/fdfb71f5077760e7ec5ca5ca03085eda.jpg)
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Last Update: Thu, 25 Feb 16