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# Find the inverse of the following matrices if it exists using elementary operations $\begin{bmatrix}2 & -6\\1 & -2\end{bmatrix}$

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## 1 Answer

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Toolbox:
• In order to find the inverse using column elementary transformation we write as A=AI
$\begin{bmatrix}2 & -6\\1 & -2\end{bmatrix}=A\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$
Step 1: Apply $C_1\rightarrow \frac{1}{2}C_1$
$\begin{bmatrix}1 & -6\\\frac{1}{2} & -2\end{bmatrix}=A\begin{bmatrix}\frac{1}{2} & 0\\0 & 1\end{bmatrix}$
Step 2: Apply $C_2\rightarrow C_2+6C_1$
$\begin{bmatrix}1 & 0\\\frac{1}{2} & 1\end{bmatrix}=A\begin{bmatrix}\frac{1}{2} & 3\\0 & 1\end{bmatrix}$
Step 3: Apply $C_1\rightarrow C_1-\frac{1}{2}C_2$
$\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}=A\begin{bmatrix}-1 & 3\\\frac{-1}{2} & 1\end{bmatrix}$

Step 4: $A^{-1}=\begin{bmatrix}-1 & 3\\\frac{-1}{2} & 1\end{bmatrix}$
answered Apr 4, 2013

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