Quiz...Carilah invers Matriks berikut:[tex]\tt{B} = \begin{gathered} \left[\begin{array}{ccc}1&2&3\\.... Question from @misrokhah439

August 2022 1 19 Report

Quiz...

Carilah invers Matriks berikut:
[tex]\tt{B} = \begin{gathered} \left[\begin{array}{ccc}1&2&3\\ - 1& - 2&4\\4& - 2&0\end{array}\right] \end{gathered}[/tex]

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henriyulianto

Invers dari matriks $B$ adalah:

$\large\text{$\begin{aligned}B^{-1}=\begin{bmatrix}\bf{}^{4}\!/_{35} & \bf-{}^{3}\!/_{35} & \bf{}^{1}\!/_{5}\\\bf{}^8\!/_{35} & \bf-{}^6\!/_{35} & \bf-{}^{1}\!/_{10}\\\bf{}^1\!/_7 & \bf{}^1\!/_7 & \bf0\end{bmatrix}\end{aligned}$}$
 

Pembahasan

Invers Matriks

Diketahui

${\sf Matriks}\ B=\begin{bmatrix}1 & 2 & 3 \\-1 & -2 & 4 \\4 & -2 & 0\end{bmatrix}$

Ditanyakan

Invers dari matriks $B$

PENYELESAIAN

Penyelesaian menggunakan Operasi Baris Elementer (OBE) untuk mencari invers dari matriks B.

Catatan: $R_n$ = "row" (baris) ke- $n$
------------------------------------------------

$\Big[B\,|\,I\Big]\::\:\left[\begin{array}{ccc|ccc}1 & 2 & 3 & 1 & 0 & 0 \\-1 & -2 & 4 & 0 & 1 & 0 \\4 & -2 & 0 & 0 & 0 & 1\end{array}\right]$

$\begin{aligned}(1):\ &\boxed{\ R_2+R_1\to R_2\ }\\(2):\ &\boxed{\ R_3-4R_1\to R_3\ }\\{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 2 & 3 & 1 & 0 & 0 \\0 & 0 & 7 & 1 & 1 & 0 \\0 & -10 & -12 & -4 & 0 & 1\end{array}\right]\end{aligned}$

$\begin{aligned}(3):\ &\boxed{\ R_2\leftrightarrow R_3\quad\textsf{(tukar posisi)}\ }\\{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 2 & 3 & 1 & 0 & 0 \\0 & -10 & -12 & -4 & 0 & 1\\0 & 0 & 7 & 1 & 1 & 0\end{array}\right]\end{aligned}$

$\begin{aligned}(4):\ &\boxed{\ R_2\times\left(-\tfrac{1}{10}\right)\to R_2\ }\\{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 2 & 3 & 1 & 0 & 0 \\0 & 1 & {}^6\!/_5 & {}^2\!/_5 & 0 & -{}^{1}\!/_{10}\\0 & 0 & 7 & 1 & 1 & 0\end{array}\right]\end{aligned}$

$\begin{aligned}(5):\ &\boxed{\ R_3\times\tfrac{1}{7}\to R_3\ }\\{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 2 & 3 & 1 & 0 & 0 \\0 & 1 & {}^6\!/_5 & {}^2\!/_5 & 0 & -{}^{1}\!/_{10}\\0 & 0 & 1 & {}^1\!/_7 & {}^1\!/_7 & 0\end{array}\right]\end{aligned}$

$\begin{aligned}(6):\ &\boxed{\ R_2-\tfrac{6}{5}R_3\to R_2\ }\\{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 2 & 3 & 1 & 0 & 0 \\0 & 1 & 0 & {}^{14}\!/_{35}-{}^6\!/_{35} & -{}^6\!/_{35} & -{}^{1}\!/_{10}\\0 & 0 & 1 & {}^1\!/_7 & {}^1\!/_7 & 0\end{array}\right]\\{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 2 & 3 & 1 & 0 & 0 \\0 & 1 & 0 & {}^8\!/_{35} & -{}^6\!/_{35} & -{}^{1}\!/_{10}\\0 & 0 & 1 & {}^1\!/_7 & {}^1\!/_7 & 0\end{array}\right]\end{aligned}$

$\begin{aligned}(7):\ &\boxed{\ R_1-3R_3\to R_1\ }\\{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 2 & 0 & 1-{}^3\!/_7 & -{}^3\!/_7 & 0 \\0 & 1 & 0 & {}^8\!/_{35} & -{}^6\!/_{35} & -{}^{1}\!/_{10}\\0 & 0 & 1 & {}^1\!/_7 & {}^1\!/_7 & 0\end{array}\right]\\{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 2 & 0 & {}^4\!/_7 & -{}^3\!/_7 & 0 \\0 & 1 & 0 & {}^8\!/_{35} & -{}^6\!/_{35} & -{}^{1}\!/_{10}\\0 & 0 & 1 & {}^1\!/_7 & {}^1\!/_7 & 0\end{array}\right]\end{aligned}$

$\begin{aligned}(8):\ &\boxed{\ R_1-2R_2\to R_1\ }\\{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 0 & 0 & {}^{20}\!/_{35}-{}^{16}\!/_{35} & -{}^{15}\!/_{35}+{}^{12}\!/_{35} & {}^{2}\!/_{10} \\0 & 1 & 0 & {}^8\!/_{35} & -{}^6\!/_{35} & -{}^{1}\!/_{10}\\0 & 0 & 1 & {}^1\!/_7 & {}^1\!/_7 & 0\end{array}\right]\\\end{aligned}$
$.\ \:{\Rightarrow\ }&\left[\begin{array}{ccc|ccc}1 & 0 & 0 & {}^{4}\!/_{35} & -{}^{3}\!/_{35} & {}^{1}\!/_{5} \\0 & 1 & 0 & {}^8\!/_{35} & -{}^6\!/_{35} & -{}^{1}\!/_{10}\\0 & 0 & 1 & {}^1\!/_7 & {}^1\!/_7 & 0\end{array}\right]$

$\begin{aligned}\Big[I\,|\,B^{-1}\Big]\::\:\left[\begin{array}{ccc|ccc}1 & 0 & 0 & \bf{}^{4}\!/_{35} & \bf-{}^{3}\!/_{35} & \bf{}^{1}\!/_{5} \\0 & 1 & 0 & \bf{}^8\!/_{35} & \bf-{}^6\!/_{35} & \bf-{}^{1}\!/_{10}\\0 & 0 & 1 & \bf{}^1\!/_7 & \bf{}^1\!/_7 & \bf0\end{array}\right]\end{aligned}$
 

KESIMPULAN

$\large\text{$\begin{aligned}\therefore\ B^{-1}=\begin{bmatrix}\bf{}^{4}\!/_{35} & \bf-{}^{3}\!/_{35} & \bf{}^{1}\!/_{5}\\\bf{}^8\!/_{35} & \bf-{}^6\!/_{35} & \bf-{}^{1}\!/_{10}\\\bf{}^1\!/_7 & \bf{}^1\!/_7 & \bf0\end{bmatrix}\end{aligned}$}$