Question

By using elementary transformation find the inverse of the matrix $A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$.

Answer 1

Given matrix is $A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$

Now we will use elementary transformation to find the inverse of given matrix

First step is to write the identity matrix at the right of given matrix

$⟹\left[\begin{array}{cccc}1& 2& 1& 0\\ 2& 1& 0& 1\end{array}\right]$

Now, we have to start applying elementary transformations until we get Identity matrix on left side.

$R_1\to R_1-2R_2$

$⟹\left[\begin{array}{cccc}-3& 0& 1& -2\\ 2& 1& 0& 1\end{array}\right]$

$R_2\to 3R_2+2R_1$

$⟹\left[\begin{array}{cccc}-3& 0& 1& -2\\ 0& 3& 2& -1\end{array}\right]$

$R_1\to \frac{R_1}{-3}$ and $R_2\to \frac{R_2}{3}$

$⟹\left[\begin{array}{cccc}1& 0& -\frac{1}{3}& \frac{2}{3}\\ 0& 1& \frac{2}{3}& -\frac{1}{3}\end{array}\right]$

Therefore the inverse of given matrix is $\left[\begin{array}{cc}-\frac{1}{3}& \frac{2}{3}\\ \frac{2}{3}& -\frac{1}{3}\end{array}\right]$