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Soal UNBM Matematika IPA tentang Matriks

Topik Bahasan

 Soal UNBK Matematika IPA 2018 

Diketahui matriks $A=\begin{pmatrix}
2 & 1\\ 4 & -1
\end{pmatrix}$ dan $B=\begin{pmatrix}
4 & -1\\ 1 & 1
\end{pmatrix}$. Jika $C=AB$, invers matriks $C$ adalah $C^{-1}=\cdots$
$\begin{align}
(A)\ & \begin{pmatrix}
\frac{1}{6} & -\frac{1}{30} \\ \frac{1}{2} & -\frac{3}{10}
\end{pmatrix} \\ (B)\ & \begin{pmatrix}
-\frac{1}{6} & \frac{1}{2} \\ -\frac{1}{30} & -\frac{3}{10}
\end{pmatrix} \\ (C)\ & \begin{pmatrix}
\frac{1}{6} & -\frac{1}{2} \\ -\frac{1}{30} & -\frac{3}{10}
\end{pmatrix} \\ (D)\ & \begin{pmatrix}
\frac{1}{6} & -\frac{1}{30} \\ -\frac{1}{2} & \frac{3}{10}
\end{pmatrix} \\ (E)\ & \begin{pmatrix}
-\frac{1}{6} & -\frac{1}{2} \\ -\frac{1}{30} & -\frac{3}{10}
\end{pmatrix}
\end{align}$


Pembahasan:

$C=AB$
$C=\begin{pmatrix}
2 & 1\\ 4 & -1
\end{pmatrix} \begin{pmatrix}
4 & -1\\ 1 & 1
\end{pmatrix}$
$C=\begin{pmatrix}
9 & -1\\ 15 & -5
\end{pmatrix}$

$C^{-1}=\frac{1}{ad-bc}\begin{pmatrix}
d & -b\\ -c & a
\end{pmatrix}$
$C^{-1}=\frac{1}{(9)(-5)-(15)(-1)}\begin{pmatrix}
-5 & 1\\ -15 & 9
\end{pmatrix}$
$C^{-1}=\frac{1}{-30}\begin{pmatrix}
-5 & 1\\ -15 & 9
\end{pmatrix}$
$C^{-1}= \begin{pmatrix}
\frac{1}{6} & -\frac{1}{30} \\ \frac{1}{2} & -\frac{3}{10}
\end{pmatrix}$

$\therefore$ Pilihan yang sesuai adalah $(A)\ \begin{pmatrix}
\frac{1}{6} & -\frac{1}{30} \\ \frac{1}{2} & -\frac{3}{10}

\end{pmatrix}$ 

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