# Using elementary transformations, find the inverse of each of the matrices, if it exists in Exercises 1 to 17.    Q8     $\begin{bmatrix} 4 &5 \\ 3 & 4 \end{bmatrix}$

S seema garhwal

$A=\begin{bmatrix} 4 &5 \\ 3 & 4 \end{bmatrix}$

$A=IA$

$\Rightarrow$          $\begin{bmatrix} 4 &5 \\ 3 & 4 \end{bmatrix}$ $= \begin{bmatrix}1&0\\0&1 \end{bmatrix}A$

$R_1\rightarrow R_1-R_2$

$\Rightarrow$          $\begin{bmatrix} 1 &1 \\ 3 & 4 \end{bmatrix}$ $= \begin{bmatrix}1&-1\\0&1 \end{bmatrix}A$

$\Rightarrow$          $R_2\rightarrow R_2-3R_1$

$\Rightarrow$          $\begin{bmatrix}1&1\\0&1 \end{bmatrix}$ $= \begin{bmatrix}1&-1\\-3&4 \end{bmatrix}A$

$R_1\rightarrow R_1-R_2$

$\Rightarrow$          $\begin{bmatrix}1&0\\0&1 \end{bmatrix}$ $= \begin{bmatrix}4&-5\\-3&4 \end{bmatrix}A$

Thus using elementary transformation inverse of A is obtained as

$\therefore A^{-1}=$$\begin{bmatrix}4&-5\\-3&4 \end{bmatrix}$.

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