Using elementary transformations,find the inv | vedclass

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(A) Let $A = \begin{bmatrix} 1 & 2 & 1 \ 3 & 2 & 3 \ 1 & 1 & 2 \end{bmatrix}$. We write $A = IA$: $\begin{bmatrix} 1 & 2 & 1 \ 3 & 2 & 3 \ 1 & 1 &

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$\begin{bmatrix} -1/4 & 3/4 & -1 \ 3/4 & -1/4 & 0 \ -1/4 & -1/4 & 1 \end{bmatrix}$

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(A) Let $A = \begin{bmatrix} 1 & 2 & 1 \ 3 & 2 & 3 \ 1 & 1 & 2 \end{bmatrix}$. We write $A = IA$: $\begin{bmatrix} 1 & 2 & 1 \ 3 & 2 & 3 \ 1 & 1 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} A$ Applying $R_2 	o R_2 - 3R_1$ and $R_3 	o R_3 - R_1$: $\begin{bmatrix} 1 & 2 & 1 \ 0 & -4 & 0 \ 0 & -1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ -3 & 1 & 0 \ -1 & 0 & 1 \end{bmatrix} A$ Applying $R_2 	o R_2 / (-4)$: $\begin{bmatrix} 1 & 2 & 1 \ 0 & 1 & 0 \ 0 & -1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 3/4 & -1/4 & 0 \ -1 & 0 & 1 \end{bmatrix} A$ Applying $R_1 	o R_1 - 2R_2$ and $R_3 	o R_3 + R_2$: $\begin{bmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} -1/2 & 1/2 & 0 \ 3/4 & -1/4 & 0 \ -1/4 & -1/4 & 1 \end{bmatrix} A$ Applying $R_1 	o R_1 - R_3$: $\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} -1/4 & 3/4 & -1 \ 3/4 & -1/4 & 0 \ -1/4 & -1/4 & 1 \end{bmatrix} A$ Thus,$A^{-1} = \begin{bmatrix} -1/4 & 3/4 & -1 \ 3/4 & -1/4 & 0 \ -1/4 & -1/4 & 1 \end{bmatrix}$.

Using elementary transformations,find the inverse of the following matrix,if it exists: $\begin{bmatrix} 1 & 2 & 1 \\ 3 & 2 & 3 \\ 1 & 1 & 2 \end{bmatrix}$

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Let $A$ be a square matrix of order $3$. Choose the correct option regarding the following statements:
$I$. There exists a matrix $B$ of order $3$ such that $AB = I_3$
$II$. There exists a matrix $C$ of order $3$ such that $CA = I_3$
$III$. $A$ is invertible

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