Find the inverse of each of the matrices, if it exists
Question:

Find the inverse of each of the matrices, if it exists

$\left[\begin{array}{ll}2 & -6 \\ 1 & -2\end{array}\right]$

 

Solution:

Let $A=\left[\begin{array}{ll}2 & -6 \\ 1 & -2\end{array}\right]$

We know that A = AI

$\therefore\left[\begin{array}{ll}2 & -6 \\ 1 & -2\end{array}\right]=A\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]=A\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right] \quad\left(\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}+3 \mathrm{C}_{1}\right)$

$\Rightarrow\left[\begin{array}{ll}2 & 0 \\ 0 & 1\end{array}\right]=A\left[\begin{array}{ll}-2 & 3 \\ -1 & 1\end{array}\right] \quad\left(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{2}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=A\left[\begin{array}{cc}-1 & 3 \\ -\frac{1}{2} & 1\end{array}\right] \quad\left(\mathrm{C}_{1} \rightarrow \frac{1}{2} \mathrm{C}_{1}\right)$

$\therefore A^{-1}=\left[\begin{array}{cc}-1 & 3 \\ -\frac{1}{2} & 1\end{array}\right]$

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