Lanjutan 4 Materi Matriks (Matematika Wajib Kelas XI)

$\color{blue}\textrm{F. Invers Matriks ordo 2x2}$

Perhatikanlah kembali materi sebelumnya berkaitan determinan matriks 2x2, yaitu

$\color{blue}\begin{array}{|c|}\hline \begin{aligned}&\color{black}\textrm{Jika matriks}\: \: \color{red}A=\begin{pmatrix} a & b\\ c & d \end{pmatrix}\\ &\color{black}\textrm{maka determinan matriks}\: \: \color{red}A\\ &\textrm{ditentukan dengan}\\ &det\: \: \color{red}A=\begin{vmatrix} a & b\\ c & d \end{vmatrix}=ad-bc \end{aligned}\\\hline \end{array}$

Jika  $\color{blue}det\: \: \color{red}A$  bernilai tidak sama dengan nol, maka invers matriks ordo 2x2 yang selanjutnya dilambangkan dengan  $\color{red}A^{\color{black}-1}$  dapat ditentukan dengan formula:

$\LARGE\color{blue}\boxed{\color{red}A^{\color{black}-1}=\color{blue}\frac{1}{\color{red}ad-bc}\begin{pmatrix} d & -b\\ -c & a \end{pmatrix}}$

$\begin{aligned}&\textbf{Sebagai}\: \: \color{blue}\textrm{CONTOH}\\ &\textrm{Diketahui sebuah matrik ordo}\: \: 2x2,\: \: \textrm{yaitu}:\\ &\color{red}A=\color{black}\begin{pmatrix} -3 & 2\\ -1 & 4 \end{pmatrix},\: \: \textrm{maka}\: \: \color{red}A^{^{\color{black}-1}}\: \: \color{black}\textrm{adalah}:\\ &\color{red}A^{^{\color{black}-1}}=\displaystyle \frac{1}{\color{black}\begin{vmatrix} \color{red}-3 & 2\\ -1 & \color{red}4 \end{vmatrix}}\color{blue}\begin{pmatrix} 4 & -2\\ 1 & -3 \end{pmatrix}\\ &\: \qquad =\displaystyle \frac{1}{\color{red}12\color{black}-(-2)}\color{blue}\begin{pmatrix} 4 & -2\\ 1 & -3 \end{pmatrix}\\ &\: \qquad=\displaystyle \frac{1}{14}\color{blue}\begin{pmatrix} 4 & -2\\ 1 & -3 \end{pmatrix}\\ &\: \qquad =\color{blue}\begin{pmatrix} \displaystyle \frac{4}{14} & \displaystyle \frac{-2}{14}\\ \displaystyle \frac{1}{14} & \displaystyle \frac{-3}{14} \end{pmatrix}\\ &\: \qquad=\color{blue}\begin{pmatrix} \displaystyle \frac{2}{7} & -\displaystyle \frac{1}{7}\\ \displaystyle \frac{1}{14} & -\displaystyle \frac{3}{14} \end{pmatrix} \end{aligned}$

$\LARGE\color{purple}\fbox{CONTOH SOAL}$

$\begin{array}{ll}\\ 1.&\textrm{Tentukanlah invers matriks berikut}\\ &\textrm{a}.\quad B=\begin{pmatrix} 5 & -3\\ 4 & -2 \end{pmatrix}\\ &\textrm{b}.\quad C=\begin{pmatrix} -3 & -5\\ 6 & 9 \end{pmatrix}\\ &\textrm{c}.\quad P=\begin{pmatrix} -1 & 2\\ -3 & 6 \end{pmatrix}\\ &\textrm{d}.\quad Q=\begin{pmatrix} 6 & 9\\ 2 & 3 \end{pmatrix}\\\\ &\textrm{Jawab}:\: \color{purple}\textrm{yang dibahas poin a saja}\\ &B^{-1}=\displaystyle \frac{1}{det\: B}\begin{pmatrix} -2 & 3\\ -4 & 5 \end{pmatrix}\\ &\qquad=\displaystyle \frac{1}{-10-(-12)}\begin{pmatrix} -2 & 3\\ -4 & 5 \end{pmatrix}=\displaystyle \frac{1}{2}\begin{pmatrix} -2 & 3\\ -4 & 5 \end{pmatrix}\\ &\qquad =\color{blue}\begin{pmatrix} -1 & \displaystyle \frac{3}{2}\\ -2 & \displaystyle \frac{5}{2} \end{pmatrix}\\ &\color{purple}\textrm{b. Silahkan dicoba sendiri}\\ &\color{purple}\textrm{c. Silahkan dicoba sendiri}\\ &\color{purple}\textrm{d. Silahkan dicoba sendiri} \end{array}$

$\begin{array}{ll}\\ 2.&\textrm{Diketahui matriks}\\ & E=\begin{pmatrix} 4 & 2\\ 5 & 3 \end{pmatrix}\\ &\textrm{a}.\quad \textrm{Tentukanlah}\\ &\qquad (\textrm{i})\: \: E^{-1}\qquad\qquad\qquad (\textrm{iii})\: \: \left ( E^{-1} \right )^{t}\\ &\qquad (\textrm{i})\: \: E^{t}\qquad\qquad\qquad\: \: \: (\textrm{iv})\: \: \left ( E^{t} \right )^{-1}\\ &\textrm{b}.\quad \textrm{Dengan menggunakan hasil-hasil}\\ &\qquad \textrm{pada a. apakah}\: \: \left ( E^{-1} \right )^{t}=\left ( E^{t} \right )^{-1}\\\\ &\textrm{Jawab}:\\ &\color{blue}\begin{aligned}\textrm{a}.\quad(\textrm{i})\: \: E^{-1}&=\displaystyle \frac{1}{2}\begin{pmatrix} 3 & -2\\ -5 & 4 \end{pmatrix}=\color{red}\begin{pmatrix} \displaystyle \frac{3}{2} & -1\\ -\displaystyle \frac{5}{2} & 2 \end{pmatrix}\\ (\textrm{ii})\: \: \: \: E^{t}&=\color{purple}\begin{pmatrix} 4 & 5\\ 2 & 3 \end{pmatrix}\\ (\textrm{iii})\: \: \: \quad&\left (E^{-1} \right )^{t}=\begin{pmatrix} \displaystyle \frac{3}{2} & -\displaystyle \frac{5}{2}\\ -1 & 2 \end{pmatrix}\\ (\textrm{iv})\: \: \: \quad&\left (E^{t} \right )^{-1}=\color{purple}\displaystyle \frac{1}{2}\begin{pmatrix} 3 & -5\\ -2 & 4 \end{pmatrix}=\begin{pmatrix} \displaystyle \frac{3}{2} & -\displaystyle \frac{5}{2}\\ -1 & 2 \end{pmatrix}\\ \textrm{b}.\quad \textrm{Dari ha}&\textrm{sil yang didapat dapat disimpulkan}\\ &\color{red}\left ( E^{-1} \right )^{t}=\left ( E^{t} \right )^{-1} \end{aligned} \end{array}$

DAFTAR PUSTAKA

1. Wirodikromo, S. 2003. Matematika 2000 untuk SMU Jilid 2 Kelas 1 Semester 2. Jakarta: ERLANGGA