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  4. Using elementary transformations, find the inverse of matrix [6&5 5&4]

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Q. Using elementary transformations, find the inverse of matrix
$\begin{bmatrix}6&5\\ 5&4\end{bmatrix}$

Matrices

Solution:

Given matrix$ A =\begin{bmatrix}6&5\\ 5&4\end{bmatrix},then A= IA$

$\Rightarrow \begin{bmatrix}6&5\\ 5&4\end{bmatrix}\begin{bmatrix}1&0\\ 0&1\end{bmatrix}A$

Applying $R_{1} \rightarrow R_{1} - R_{2}$, we get

$\begin{bmatrix}1&1\\ 5&4\end{bmatrix}=\begin{bmatrix}1&-1\\ 0&1\end{bmatrix}A$

Applying $R_{2} \rightarrow R_{2} - 5R_{1}$, we get

$\begin{bmatrix}1&1\\ 0&-1\end{bmatrix}=\begin{bmatrix}-4&5\\ -5&6\end{bmatrix}A$

Applying $R_{2} \rightarrow\left(-1\right) R_{2}$, we get

$\begin{bmatrix}1&0\\ 0&1\end{bmatrix}=\begin{bmatrix}-4&5\\ 5&-6\end{bmatrix}A$

$A^{-1}= \begin{bmatrix}-4&5\\ 5&-6\end{bmatrix}$