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

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

$\left[\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right]$

 

Solution:

Let $A=\left[\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right]$

We know that $A=I A$

$\therefore\left[\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] A$

$\Rightarrow\left[\begin{array}{rr}1 & -1 \\ 0 & 5\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ -2 & 1\end{array}\right] A \quad\left(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-2 \mathrm{R}_{1}\right)$

$\Rightarrow\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ -\frac{2}{5} & \frac{1}{5}\end{array}\right] A \quad\left(\mathrm{R}_{2} \rightarrow \frac{1}{5} \mathrm{R}_{2}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}\frac{3}{5} & \frac{1}{5} \\ -\frac{2}{5} & \frac{1}{5}\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}\right)$

$\therefore A^{-1}=\left[\begin{array}{ll}\frac{3}{5} & \frac{1}{5} \\ -\frac{2}{5} & \frac{1}{5}\end{array}\right]$

 

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