Lanjutan 2 Materi Matriks (Matematika Wajib Kelas XI)

 $\text{C. Tarnspose dan Kesamaan Dua Buah Matriks}$

$\begin{array}{|cl|}\hline 1.& \text{Transpose Matriks}\\ & \begin{array}{rl}& \text{Membentuk matriks baru dari matriks}\\ & \text{dengan cara mengubah baris matriks ke}-i\\ & \text{menjadi kolom ke}-i,\text{pada matriks baru}\\ & \text{dan demikian pula untuk kolomnya}.\text{Jika}\\ & \text{matriks pertama adalah A maka matriks}\\ & \text{transposenya adalah}{\text{A}}^{\prime }\text{atau}{\text{A}}^t\end{array}\\ 2.& \text{Kesamaan Duan Buah Matriks}\\ & \begin{array}{rl}& \text{Misalkan matriks}\text{A}=\left(a_{ij}\right)\text{dan}\text{B}=\left(b_{ij}\right)\\ & \text{adalah dua buah matriks berordo sama},\\ & \text{maka matriks A dikatakan sama dengan matriks B}\\ & \text{jika elemen-elemen yang seletak sama pada}\\ & \text{kedua matriks tersebut bernilai sama}\\ & \end{array}\\ \hline\end{array}$

$\begin{array}{|l|}\hline \text{Berikut contoh transpose}\\ \text{A}={\displaystyle \left(\begin{array}{cc}1& 2\\ -7& 0\\ 5& 4\end{array}\right)\Rightarrow {\text{A}}^t=\left(\begin{array}{ccc}1& -7& 5\\ 2& 0& 4\end{array}\right)}\\ \text{DAn berikut contoh kesamaan dua matriks}\\ \text{A}={\displaystyle \left(\begin{array}{cc}1& 2\\ -7& 0\\ 5& 4\end{array}\right),\text{D}={\displaystyle \left(\begin{array}{cc}1& 2\\ -7& 0\\ 5& 4\end{array}\right),\Rightarrow \text{A}=\text{D}}}\\ \hline\end{array}$

 $\text{D. Operasi Matriks}$

$\begin{array}{|clll|}\hline \text{No}& \text{Operasi}& \text{Ketentuan}& \text{Contoh}\\ & \text{Matriks}& & \\ 1& \text{Penjumlahan}\mathrm{\&}& \text{ordo sama}& A=\left(\begin{array}{c}1\\ 2\end{array}\right),B=\left(\begin{array}{c}8\\ 9\end{array}\right),\text{maka}\\ 2& \text{Pengurangan}& \text{ordo sama}& A+B=\left(\begin{array}{c}1+8\\ 2+9\end{array}\right)=\left(\begin{array}{c}9\\ 11\end{array}\right)\\ 3& \text{Perkalian}& \text{Dengan}& k\left(\begin{array}{cc}p& q\\ r& s\end{array}\right)=\left(\begin{array}{cc}kp& kq\\ kr& ks\end{array}\right)\\ & \text{Skalar}& \text{mengalikan}& \\ & & \text{ke setiap elemen}& \\ 4& \text{Perkalian}& \begin{array}{rl}& \text{Dua matriks }\\ & \text{dapat dikalikan }\\ & \text{jika}\\ & \text{banyaknya kolom}\\ & \text{matriks pertama}\\ & \text{sama dengan}\\ & \text{banyaknya baris}\\ & \text{matriks kedua}\end{array}& \begin{array}{rl}E=& \left(\begin{array}{cc}1& 2\\ 3& -1\end{array}\right),F=\left(\begin{array}{c}5\\ 0\end{array}\right),\\ & \text{maka}\\ E& \times F\\ & ={\left(\begin{array}{cc}1& 2\\ 3& -1\end{array}\right)}_{2\times 2}\times {\left(\begin{array}{c}5\\ 0\end{array}\right)}_{2\times 1}\\ & \text{syarat memenuhi yaitu}:\\ & \text{kolom matriks 1}\\ & =\text{baris matriks 2}\\ & \text{dan hasilnya adalah }\\ & \text{matriks baru}\\ & \text{dengan ordo }\\ & \text{banyak baris matriks 1}\\ & \text{kali banyak}\\ & \text{kolom matriks 2}\\ & \text{Dan aturan perkaliannya }\\ & \text{adalah}\\ & \text{elemen baris matriks 1 kali}\\ & \text{elemen kolom matriks 2}\\ & \text{sehingga}\\ & ={\left(\begin{array}{c}1(5)+2(0)\\ 3(5)+-1(0)\end{array}\right)}_{2\times 1}\\ & =\left(\begin{array}{c}5+0\\ 15-0\end{array}\right)={\left(\begin{array}{c}5\\ 15\end{array}\right)}_{2\times 1}\end{array}\\ \hline\end{array}$

$\boxed{\text{CONTOH SOAL}}$

$\begin{array}{l}\\ \bullet & \text{Penjumlahan}\\ & \begin{array}{rl}& \left(\begin{array}{cc}1& 2\\ 3& -4\end{array}\right)+\left(\begin{array}{cc}5& 6\\ 7& -8\end{array}\right)\\ & =\left(\begin{array}{cc}1+5& 2+6\\ 3+7& (-4)+(-8)\end{array}\right)\\ & =\left(\begin{array}{cc}6& 8\\ 10& -12\end{array}\right)\end{array}\\ \bullet & \text{Lawan suatu matriks}\\ & \begin{array}{rl}& \text{Jika}A=\left(\begin{array}{cc}1& 2\\ 3& -4\end{array}\right),\\ & \text{maka lawan matriks A adalah -A,}\\ & \text{Sehingga}-A=\left(\begin{array}{cc}-1& -2\\ -3& 4\end{array}\right)\end{array}\\ \bullet & \text{Pengurangan}\\ & \begin{array}{rl}& \left(\begin{array}{cc}1& 2\\ 3& -4\end{array}\right)-\left(\begin{array}{cc}5& 6\\ 7& -8\end{array}\right)\\ & =\left(\begin{array}{cc}1-5& 2-6\\ 3-7& (-4)-(-8)\end{array}\right)\\ & =\left(\begin{array}{cc}-4& -4\\ -4& 4\end{array}\right)\end{array}\\ \bullet & \text{Perkalian}\\ & \begin{array}{rl}& (1)\text{Perkalian suatu matriks dengan skalar}k\\ & 2\times \left(\begin{array}{cc}1& 2\\ 3& -4\end{array}\right)=\left(\begin{array}{cc}2\times 1& 2\times 2\\ 2\times 3& 2\times (-4)\end{array}\right)\\ & =\left(\begin{array}{cc}2& 4\\ 6& -8\end{array}\right)\end{array}\\ & \begin{array}{rl}& (2)\text{Perkalian antara dua buah matriks}\\ & \left(\begin{array}{cc}1& 2\\ 3& -4\end{array}\right)\times \left(\begin{array}{cc}5& 6\\ 7& -8\end{array}\right)\\ & \text{perhatikan syarat memenuhi}\\ & =\left(\begin{array}{cc}1\times 5+2\times 7& 1\times 6+2\times (-8)\\ 3\times 5+(-4)\times 7& 3\times 6+(-4)\times (-8)\end{array}\right)\\ & =\left(\begin{array}{cc}5+14& 6+(-16)\\ 15+(-28)& 18+32\end{array}\right)\\ & =\left(\begin{array}{cc}19& -10\\ -13& 50\end{array}\right)\end{array}\end{array}$