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Example Find the inverse matrix of the followings:

$\displaystyle A = \left(\begin{array}{rrr} 1&2&3\\ 2&-1&1\\ 4&3&2 \end{array}\right)$

Answer

$\displaystyle [A:I]$ $\displaystyle =$ $\displaystyle \left(\begin{array}{rrrrrr} 1&2&3&1&0&0\\ 2&-1&1&0&1&0\\ 4&3&2&... ...y}{rrrrrr} 1&2&3&1&0&0\\ 0&-5&-5&-2&1&0\\ 0&-5&-10&-4&0&1 \end{array}\right )$  
  $\displaystyle \longrightarrow$ $\displaystyle \left(\begin{array}{rrrrrr} 1&2&3&1&\!\!0&\!\!0\\ 0&1&1&\frac{2}... ...rac{1}{5}&\!\!0\\ 0&1&2&\frac{4}{5}&\!\!0&\!\!-\frac{1}{5} \end{array}\right )$  
  $\displaystyle \longrightarrow$ $\displaystyle \left(\begin{array}{rrrrrr} 1&0&0&-\frac{1}{5}&\frac{1}{5}&\frac{... ... 0&0&1&\frac{2}{5}&\frac{1}{5}&-\frac{1}{5} \end{array}\right ) = [I:A^{-1}] .$  

Thus, $A$ is regular.

$\displaystyle A^{-1} = \left(\begin{array}{rrr} -\frac{1}{5}&\frac{1}{5}&\frac{... ...{2}{5}&\frac{1}{5}\\ \frac{2}{5}&\frac{1}{5}&-\frac{1}{5} \end{array}\right ).$