Question
Find the inverse of the following matrices by using elementry row transformation:
$\begin{bmatrix}2 & 5 \\ 1 & 3 \end{bmatrix}$
$\begin{bmatrix}2 & 5 \\ 1 & 3 \end{bmatrix}$
We know A = IA
$\Rightarrow\begin{bmatrix}2 & 5 \\ 1 & 3 \end{bmatrix}=\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}\text{A}$
$\Rightarrow\begin{bmatrix}2-1 & 5-3 \\ 1 & 3 \end{bmatrix}=\begin{bmatrix}1-0 & 0-1 \\ 0 & 1 \end{bmatrix}\text{A}$
[Applying R1 → R1 - R2]
$\Rightarrow\begin{bmatrix}1 & 2 \\ 1 & 3 \end{bmatrix}=\begin{bmatrix}1 & -1 \\ 0 & 1 \end{bmatrix}\text{A}$
$\Rightarrow\begin{bmatrix} 1 & 2 \\ 1-1 & 3-2 \end{bmatrix}=\begin{bmatrix} 1 & -1 \\ 0-1 & 1+1 \end{bmatrix}\text{A}$
[Applying R2 → R1 - R2]
$\Rightarrow\begin{bmatrix}1 & 2 \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}\text{A}$
$\Rightarrow\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 1+2 & -1-4 \\ -1 & 2 \end{bmatrix}\text{A}$
[Applying R1 → R1 - 2R2]
$\Rightarrow\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}=\begin{bmatrix}3 & -5 \\ -1 &2 \end{bmatrix}\text{A}$
$\Rightarrow\text{A}^{-1}=\begin{bmatrix}3 & -5 \\ -1 & 2 \end{bmatrix}$
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