Using elementary row operations, find the inverse of the followin… - Vidyadip

Question

Using elementary row operations, find the inverse of the following matrix:$\begin{pmatrix} 2 & 5 \\ 1 & 3 \\ \end{pmatrix}.$

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Answer

Here $A = \begin{pmatrix} 2 & 5 \\ 1 & 3 \\ \end{pmatrix}$
Writing $A= IA \Rightarrow$ $\begin{vmatrix} 2 & 5 \\ 1 & 3 \\ \end{vmatrix}=\begin{vmatrix} 1 & 0 \\ 0 & 1 \\ \end{vmatrix}\text{A}$
Applying $R_1 \rightarrow R_{1 }- R_{2, }$ we get
$\begin{vmatrix} 1 & 2 \\ 1 & 3 \\ \end{vmatrix}=\begin{vmatrix} 1 & -1 \\ 0 & 1 \\ \end{vmatrix}\text{A}$
Applying $R_2 \rightarrow R_2-R_1$, we get $\begin{bmatrix} 1 & 2 \\ 1 & 3 \\ \end{bmatrix}$= $\begin{bmatrix} 1 & -1 \\ -1 & 2 \\ \end{bmatrix}\text{A}$
Applying $R_1 \rightarrow R_{1 }- 2R_{2, }$ we get $\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$= $\begin{bmatrix} 3 & -5 \\ -1 & 2 \\ \end{bmatrix}\text{A}$
$\Rightarrow\text{A}^{-1}=\begin{bmatrix} 3 & -5 \\ -1 & 2 \\ \end{bmatrix}$.

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