Question

If A=[ 1 2 3 ] [ 1 1 5] [2 4 7] inverse by elementary column transformation .​

Answer 1

$\begin{gathered}\sf A=\left[\begin{array}{ccc}1&2&3\\1&1&5\\2&4&7\end{array}\right]\end{gathered}$

$\displaystyle\longrightarrow \tt \: Let \: A \: = \: A \: I$

$\begin{gathered}\sf\left[\begin{array}{ccc} 1\: & 2& 3\: \\1&1&5\\2&4&7\end{array}\right]\end{gathered} \: =A \begin{gathered}\sf\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]\end{gathered}$

$\tt \:  OP \: C_2\displaystyle\longrightarrow \: C_2 - 2C_1$

$\begin{gathered}\sf\left[\begin{array}{ccc}1&0&3\\1& - 1&5\\2&0&7\end{array}\right]\end{gathered} = A \begin{gathered}\sf\left[\begin{array}{ccc}1& - 2&0\\0&1&0\\0&0&1\end{array}\right]\end{gathered}$

$\tt \:  OP \: C_3\displaystyle\longrightarrow \: C_3 - 3C_1$

$\begin{gathered}\sf\left[\begin{array}{ccc}1&0&0\\1& - 1&2\\2&0&1\end{array}\right]\end{gathered} = A \begin{gathered}\sf\left[\begin{array}{ccc}1& - 2& - 3\\0&1&0\\0&0&1\end{array}\right]\end{gathered}$

$\tt \:  OP \: C_2\displaystyle\longrightarrow \: ( - 1) C_2$

$\begin{gathered}\sf\left[\begin{array}{ccc}1&0&0\\1& 1&2\\2&0&1\end{array}\right]\end{gathered} = A \begin{gathered}\sf\left[\begin{array}{ccc}1& 2& - 3\\0& - 1&0\\0&0&1\end{array}\right]\end{gathered}$

$\tt \:  OP \: C_1\displaystyle\longrightarrow \: C_1 - C_2 \: \tt \: and \: \tt \:  OP \: C_3\displaystyle\longrightarrow \: C_3 - 2C_2$

$\begin{gathered}\sf\left[\begin{array}{ccc}1&0&0\\0& 1&0\\2&0&1\end{array}\right]\end{gathered} = A \begin{gathered}\sf\left[\begin{array}{ccc} - 1& 2& - 7\\1& - 1&2\\0&0&1\end{array}\right]\end{gathered}$

$\tt \:  OP \: C_1\displaystyle\longrightarrow \: C_1 - 2C_3$

$\begin{gathered}\sf\left[\begin{array}{ccc}1&0&0\\0& 1&0\\0&0&1\end{array}\right]\end{gathered} = A \begin{gathered}\sf\left[\begin{array}{ccc} 13& 2& - 7\\ - 3& - 1&2\\ - 2&0&1\end{array}\right]\end{gathered}$

$\bf\implies \: {A}^{ - 1} = \begin{gathered}\sf\left[\begin{array}{ccc} 13& 2& - 7\\ - 3& - 1&2\\ - 2&0&1\end{array}\right]\end{gathered}$

Answer 2

Answer:

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